# How to Calculate the Sides and Angles of Triangles

Updated on October 26, 2019 Eugene is a qualified control/instrumentation engineer Bsc (Eng) and has worked as a developer of electronics & software for SCADA systems.

## Trigonometry and the Basics of Triangles

In this tutorial, you'll learn about trigonometry which is a branch of mathematics that covers the relationship between the sides and angles of triangles. We'll cover the basic facts about triangles first, then learn about Pythagoras' theorem, the sine rule, the cosine rule and how to use them to calculate all the angles and side lengths of triangles when you only know some of the angles or side lengths. You'll also discover different methods of working out the area of a triangle.

## What Is a Triangle?

By definition, a triangle is a polygon with three sides.

Polygons are plane (flat, two-dimensional) shapes with several straight sides. Other examples include squares, pentagons, hexagons and octagons. The name originates from the Greek polús meaning "many" and gōnía meaning "corner" or "angle." So polygon means "many corners." A triangle is the simplest polygon, having only three sides.

The most basic fact about triangles is that all the angles add up to a total of 180 degrees. The angle between the sides can be anything from greater than 0 to less than 180 degrees. The angles can't be 0 or 180 degrees, because the triangles would become straight lines. (These are called degenerate triangles).

Degrees can be written using the symbol º. So, 45º means 45 degrees.

Triangles come in many shapes and sizes according to the angles of their corners. Some triangles, called similar triangles, have the same angles but different side lengths. This changes the ratio of the triangle, making it bigger or smaller, without changing the degree of its three angles.

Below, we will examine the many ways to discover the side lengths and angles of a triangle. Angles of a triangle range from 0 to less than 180 degrees. | Source No matter what the shape or size of a triangle, the sum of the 3 angles is 180 degrees. | Source Similar triangles. | Source

## What is the Triangle Inequality Theorem?

This states that the sum of any two sides of a triangle must be greater than or equal to the remaining side.

## What Are the Different Types of Triangles?

Before we learn how to work out the sides and angles of a triangle, it's important to know the names of the different types of triangles. The classification of a triangle depends on two factors:

• The length of a triangle's sides
• The angles of a triangle's corners

Below is a graphic and table listing the different types of triangles along with a description of what makes them unique.

## Types of Triangles

You can classify a triangle either by side length or internal angle.

## By Lengths of Sides

Type of Triangle
Description
Isosceles
An isosceles triangle has two sides of equal length, and one side that is either longer or shorter than the equal sides. Angle has no bearing on this triangle type.
Equilateral
All sides and angles are equal in length and degree.
Scalene
All sides and angles are of different lengths and degrees.
Types of triangles by length of sides.

## By Internal Angle

Type of Triangle
Description
Right (right angled)
One angle is 90 degrees.
Acute
Each of the three angles measure less the 90 degrees.
Obtuse
One angle is greater than 90 degrees.
Types of triangles by angle.

## Using the Greek Alphabet for Equations

Another topic that we'll briefly cover before we delve into the mathematics of solving triangles is the Greek alphabet.

In science, mathematics, and engineering many of the 24 characters of the Greek alphabet are borrowed for use in diagrams and for describing certain quantities. For example, the characters θ (theta) and φ (phi) are often used for representing angles.

You may have also seen the character μ (mu) represent micro as in micrograms μg or micrometers μm. The capital letter Ω (omega) is the symbol for ohms in electrical engineering. And, of course, π (pi) is the ratio of the circumference to the diameter of a circle.

## How Do You Find the Sides and Angles of a Triangle?

There are many methods available when it comes to discovering the sides and angles of a triangle. To find the length or angle of a triangle, one can use formulas, mathematical rules, or the knowledge that the angles of all triangles add up to 180 degrees.

Tools to Discover the Sides and Angles of a Triangle

• Pythagoras' theorem
• Sine rule
• Cosine rule
• The fact that all angles add up to 180 degrees

## Pythagoras' Theorem (The Pythagorean Theorem)

Pythagoras' theorem uses trigonometry to discover the longest side (hypotenuse) of a right triangle (right angled triangle in British English). It states that for a right triangle:

The square on the hypotenuse equals the sum of the squares on the other two sides.

Written as a formula, Pythagoras' theorem is as follows:

c² = a² + b²

c = √(a² + b²)

The hypotenuse is the longest side of a right triangle, and is thus located opposite the right angle.

So, if you know the lengths of two sides, all you have to do is square the two lengths, add the result, then take the square root of the sum to get the length of the hypotenuse.

### Example Problem Using the Pythagorean Theorem

The sides of a triangle are 3 and 4 units long. What is the length of the hypotenuse?

Call the sides a, b, and c. Side c is the hypotenuse.

a = 3
b = 4

c = Unknown

So, according to the Pythagorean theorem:

c² = a² + b²

So, c² = 3² + 4² = 9 + 16 = 25

c = √25

c = 5

## Sine, Cosine, and Tan of an Angle

A right triangle has one angle measuring 90 degrees. The side opposite this angle is known as the hypotenuse (another name for the longest side). The length of the hypotenuse can be discovered using Pythagoras' theorem, but to discover the other two sides, sine and cosine must be used. These are trigonometric functions of an angle.

In the diagram below, one of the angles is represented by the Greek letter θ. Side a is known as the "opposite" side and side b is "adjacent" to the angle θ.

The vertical lines "||" around the words below mean "length of."

sine θ = |opposite side| / |hypotenuse|

cosine θ = |adjacent side| / |hypotenuse|

Tan θ = |opposite side| / |adjacent side|

Sine and cosine apply to an angle, any angle, so it's possible to have two lines meeting at a point and to evaluate sine or cos for that angle. However, sine and cosine are derived from the sides of an imaginary right triangle superimposed on the lines.

In the second diagram below, you can imagine a right angled triangle superimposed on the purple triangle, from which the opposite, adjacent, hypotenuse sides can be determined.

Over a range 0 to 90 degrees, sine ranges from 0 to 1, and cos ranges from 1 to 0.

Remember, sine and cosine only depend on the angle, not the size of the triangle. So if the length a changes in the diagram below when the triangle changes in size, the hypotenuse c also changes in size, but the ratio of a to c remains constant. They are similar triangles.

Sine and cosine are sometimes abbreviated to sin and cos.

### The Sine Rule

The ratio of the length of a side of a triangle to the sine of the angle opposite is constant for all three sides and angles.

So, in the diagram below:

a / sine A = b / sine B = c / sine C

Now, you can check the sine of an angle using a scientific calculator or look it up online. In the old days before scientific calculators, we had to look up the value of the sine or cos of an angle in a book of tables.

The opposite or reverse function of sine is arcsine or "inverse sine", sometimes written as sin-1. When you check the arcsine of a value, you're working out the angle which produced that value when the sine function was operated on it. So:

sin (30º) = 0.5 and sin-1(0.5) = 30º

The Sine Rule Should Be Use If ...

The length of one side and the magnitude of the angle opposite is known. Then, if any of the other remaining angles or sides are known, all the angles and sides can be worked out.

### The Cosine Rule

For a triangle with sides a, b, and c, if a and b are known and C is the included angle (the angle between the sides), C can be worked out with the cosine rule. The formula is as follows:

c2 = a2 + b2 - 2abCos C

The Cosine Rule Should Be Used If ...

1. You know the lengths of two sides of a triangle and the included angle. You can then work out the length of the remaining side using the cosine rule.
2. You know all the lengths of the sides but none of the angles.

Then, by rearranging the cosine rule equation:

C = Arccos ((a2 + b2 - c2) / 2ab)

The other angles can be worked out similarly.

Click thumbnail to view full-size    ## How to Get the Area of a Triangle

There are three methods that can be used to discover the area of a triangle.

Method 1. Using the perpendicular height

The area of a triangle can be determined by multiplying half the length of its base by the perpendicular height. Perpendicular means at right angles. But which side is the base? Well, you can use any of the three sides. Using a pencil, you can work out the area by drawing a perpendicular line from one side to the opposite corner using a set square, T-square, or protractor (or a carpenter's square if you're constructing something). Then, measure the length of the line and use the following formula to get the area:

Area = 1/2ah

"a" represents the length of the base of the triangle and "h" represents the height of the perpendicular line.

Method 2. Using side lengths and angles

The simple method above requires you to actually measure the height of a triangle. If you know the length of two of the sides and the included angle, you can work out the area analytically using sine and cosine (see diagram below).

Method 3. Use Heron's formula

All you need to know are the lengths of the three sides.

Area = √(s(s - a)(s - b)(s - c))

Where s is the semiperimeter of the triangle

s = (a + b + c)/2

## Three Ways of Working Out the Area of a Triangle

Click thumbnail to view full-size    Area of a triangle equals half the base length multiplied by the perpendicular height. | Source Area = 1/2 the product of the sides multiplied by the sine of the included angle. | Source Calculation of area using Heron's formula | Source

## How Do You Measure Angles?

You can use a protractor or a digital angle finder. These are useful for DIY and construction if you need to measure an angle between two sides, or transfer the angle to another object. You can use this as a replacement for a bevel gauge for transferring angles e.g. when marking the ends of rafters before cutting. The rules are graduated in inches and centimetres and angles can be measured to 0.1 degrees. An angle finder can be used to measure cut timber, and also as a bevel gauge to transfer angles when it's necessary to cut more pieces. | Source

## Summary

If you've made it this far, you've learned numerous helpful methods to discover different aspects of a triangle. With all this information, you may be confused as to when you should use which method. The table below should help you identify which rule to use depending on the parameters you have been given.

## Find the Angles and Sides of a Triangle - Which Rule Do I Use?

Known Parameters
Triangle Type
Rule to Use
Triangle is right and I know length of two sides.
SSS after Pythagoras's Theorem used
Use Pythagoras's Theorem to work out remaining side and sine rule to work out angles.
Triangle is right and I know the length of one side and one angle
AAS after third angle worked out
Use the trigonometric identities sine and cosine to work out the other sides and sum of angles (180 degrees) to work out remaining angle.
I know the length of two sides and the angle between them.
SAS
Use the cosine rule to work out remaining side and sine rule to work out remaining angles.
I know the length of two sides and the angle opposite one of them.
SSA
Use the sine rule to work out remaining angles and side.
I know the length of one side and all three angles.
AAS
Use the sine rule to work out the remaining sides.
I know the lengths of all three sides
SSS
Use the cosine rule in reverse to work out each angle. C = Arccos ((a² + b² - c²) / 2ab)
I know the length of a side and the angle at each end
AAS
Sum of three angles is 180 degrees so remainging angle can be calculated. Use the sine rule to work out the two unknown sides
I know the length of a side and one angle

You need to know more information, either one other side or one other angle. Thes exception is if the known angle is in a rightangled triangle and not the right angle.
Summary of how to work out angles and sides of a triangle

### How Many Degrees Are There in a Triangle?

The interior angles of all triangles add up to 180 degrees.

### What Is the Hypotenuse of a Triangle?

The hypotenuse of a triangle is its longest side.

### What Do the Sides of a Triangle Add up to?

The sum of the sides of a triangle depend on the individual lengths of each side. Unlike the interior angles of a triangle, which always add up to 180 degrees

### How Do You Calculate the Area of a Triangle?

To calculate the area of a triangle, simply use the formula:

Area = 1/2ah

"a" represents the length of the base of the triangle. "h" represents its height, which is discovered by drawing a perpendicular line from the base to the peak of the triangle.

### How Do You Find the Third Side of a Triangle That Is Not Right?

If you know two sides and the angle between them, use the cosine rule and plug in the values for the sides b, c, and the angle A.

Next, solve for side a.

Then use the angle value and the sine rule to solve for angle B.

Finally, use your knowledge that the angles of all triangles add up to 180 degrees to find angle C.

### How Do You Find the Missing Side of a Right Angled Triangle?

Use the Pythagorean theorem to find the missing side of a triangle. The formula is as follows:

c² = a² + b²

c = a² + b²

### What Is the Name of a Triangle With Two Equal Sides?

A triangle with two equal sides and one side that is longer or shorter than the others is called an isosceles triangle.

### What Is the Cosine Formula?

This formula gives the square on a side opposite an angle, knowing the angle between the other two known sides. For a triangle, with sides a,b and c and angles A, B and C the three formulas are:

a2 = b2 + c2 - 2bc cos A

or

b2 = a2 + c2 - 2ac cos B

or

c2 = a2 + b2 - 2ab cos C

### How Do I Calculate the Volume of a Triangle?

Since a triangle is a plane and two-dimensional object, it is impossible to discover its volume. A triangle is flat. Thus, it has no volume.

Triangular prisms, on the other hand, are three-dimensional objects with a determinable volume. To determine the volume of a triangular prism, you must discover the area of the base of the prism, then multiply it by the height. The formula is as follows:

V = bh

In the above formula, "V" represents volume, "b" represents the area of the base of the triangular prism, and "h" represents the height of the triangular prism.

### How to Figure Out the Sides of a Triangle if I Know All the Angles?

You need to know at least one side, otherwise you can't work out the lengths of the triangle. There's no unique triangle that has all angles the same. Triangles with the same angles are similar but the ratio of sides for any two triangles is the same.

### How to Work Out the Sides of a Triangle if I know All the Sides?

Use the cosine rule in reverse.
The cosine rule states:

c2 = a2 + b2 - 2abCos C

Then, by rearranging the cosine rule equation, you can work out the angle

C = Arccos ((a2 + b2 - c2) / 2ab)

and

B = Arccos ((a2 + c2 - b2) / 2ac)

The third angle A is (180 - C - B)

Click thumbnail to view full-size   You can measure an angle with a protractor. | Source You can measure an angle with a digital angle finder. | Source

## Triangles in the Real World

A triangle is the most basic polygon and can't be pushed out of shape easily, unlike a square. If you look closely, triangles are used in the designs of many machines and structures because the shape is so strong.

The strength of the triangle lies in the fact that when any of the corners are carrying weight, the side opposite acts as a tie, undergoing tension and preventing the framework from deforming. For example, on a roof truss the horizontal ties provide strength and prevent the roof from spreading out at the eaves.

The sides of a triangle can also act as struts, but in this case they undergo compression. An example is a shelf bracket or the struts on the underside of an airplane wing or the tail wing itself.

Click thumbnail to view full-size      ## How to Implement the Cosine Rule in Excel

You can implement the cosine rule in Excel using the ACOS Excel function to evaluate arccos. This allows the included angle to be worked out, knowing all three sides of a triangle. Using the Excel ACOS function to work out an angle, knowing three sides of a triangle. ACOS returns a value in radians. | Source

## Calling All Teachers and Students

Teachers and students, would you you like to see more help guides like this one?
Please leave a suggestion in the comment section below if you have any ideas.

• How do you find the remaining sides of a triangle if you have only one angle and one side given?

You need to have more information. So either one side and the two angles at each end or two sides and the angle between them.

You can prove this to yourself by drawing out the single side and angle and seeing how you can draw as many different shaped triangles as you want.

• How do I find the value if all three sides of a scalene triangle are unknown?

If all the sides are unknown, you can't solve the triangle. You need to know at least two angles and one side, or two sides and one angle, or one side and one angle if the triangle is a right-angled triangle.

• What is the formula for finding what an equilateral triangle of side a, b and c is?

Since the triangle is equilateral, all the angles are 60 degrees. However, the length of at least one side must be known. Once you know that length, since the triangle is equilateral, you know the length of the other sides because all sides are of equal length.

• How would you solve this problem: The angle of elevation of the top of a tree from point P due west of the tree is 40 degrees. From a second point Q due east of the tree, the angle of elevation is 32 degrees. If the distance between P and Q is 200m, find the height of the tree, correct to four significant figures?

One angle is 40 degrees, the other angle is 32 degrees, therefore the third angle opposite the base PQ is 180 - (32 + 40) = 108 degrees.

You know one side of the triangle has length PQ = 200 m

A right angled triangle is formed between point P, the top of the tree and its base and also point Q, the top of the tree and its base.

The best way to solve is to find the hypotenuse of one of the triangles.

So use the triangle with vertex P.

Call the point at the top of the tree T

Call the height of the tree H

The angle formed between sides PT and QT was worked out as 108 degrees.

Using the Sine Rule, PQ / Sin(108) = PT/ Sin(32)

So for the right angled triangle we chose, PT is the hypotenuse.

Rearranging the equation above

PT = PQSin(32) / Sin(108)

Sin(40) = H / PT

So H = PTSin(40)

Substituting the value for the hypotenuse PT we calculated above gives

H = (PQSin(32) / Sin(108)) x Sin(40)

= PQSin(32)Sin(40)/Sin(108)

= 71.63 m

• How do I find the missing side of a triangle when only its height is known?

Use Pythagoras's Theorem. Add the sine, cosine and tan relationships between angles and the hypotenuse of the triangle to work out the remaining side.

0 of 8192 characters used
• AUTHOR

Eugene Brennan

7 weeks ago from Ireland

Hi Natalia,

Look at method 2 in the tutorial for finding the area of a triangle.

So the area is 1/2 the product of two sides multiplied by the sine of the angle between them.

In your question the sides are PQ and QR and the angle between them is PQR.

So area = (1/2) PQ sin PQR

Substitute for P, Q, angle PQR and the area:

14.2 = (1/2) x 7 x 5 x sin PQR

Rearrange:

sin PQR = 14.2 / ( (1/2) x 7 x 5 )

Take the arcsin of both sides. You can do all this on a calculator, but take care entering all the brackets and numbers because it's very easy to make a mistake. Make sure the calculators is set to "DEG" and use the sin ^ -1 (usually shift on sin) to work out arcsin.

I would recommend HiPer Calc as a good, free scientific calculator app for Android if you have a smartphone.

PQR = arcsin (14.2 / ( (1/2) x 7 x 5 ) ) = 54.235° = 54° 15' approx

• natalia

7 weeks ago

HI EUGENE, can you solve this problem for me and provide me with working out.

the area of triange PQR is 14.2cm squared, find angle PQR to the nearest minute, given PQ is 7cm and QR is 5cm.

• AUTHOR

Eugene Brennan

2 months ago from Ireland

Hi Pavel,

By diagonal, I presume you mean the hypotenuse.

So you can use Pythagoras' Theorem.

The square on the hypotenuse equals the sum of the squares on the other two sides.

Square the two sides and add together:

(n + 4)² + 16² = (n + 8)²

Expand out:

n² + 8n + 16 + 256 = n² + 16n + 64

Rearrange and simplify:

8n = 208

Giving n = 26

So the two sides are n + 4 = 30 cm and n + 8 = 34 cm

• Pavel

2 months ago

I have a problem about a question can you help me please?

I have a right angled triangle the bottom line is 16 cm the one on the side is n+4 and the diagonal line is n+8 can you help me find the two sides please?

• AUTHOR

Eugene Brennan

2 months ago from Ireland

Hi Carcada. You can't. You can have as many triangles as you want with exactly the same three angles. These are called similar triangles. You need to know at least the length of one side, then you can use the sine rule to work out the others.

• 2 months ago

if only the angles of each side of the triangle is given then how can we find the length of each side of the triangle?

• AUTHOR

Eugene Brennan

3 months ago from Ireland

You don't have enough information. You need to have at least one of a, c, A or C.

Sin B = 1/ sqrt 3, only gives you the angle B = (acos (1/sqrt 3)). So if a is the base, side c can be any length without knowing the other sides/angles.

• 3 months ago

I have a question. How do I find the missing sides of a triangle if I know that sin B=1/sqrt 3 and a=2

• AUTHOR

Eugene Brennan

4 months ago from Ireland

tan (ɵ) = opposite / adjacent so opposite = adjacent x tan (ɵ)

Now you know the opposite and adjacent sidfes, use Pythagoras' theorem to work out the hypotenuse.

• Phoebe

4 months ago

Hey, i have a triangle, all that is known is the adjacent, the right angle and the theta, how do i figure out the other sides,

• asaba charles

4 months ago

thanks

• Maribel Gibbs

6 months ago from Paoli, Pennsylvania

Wow, amazing! One of the best works I ever have seen here!

• Khaleel Yusuf

7 months ago

A good review of many years of wining and dining with math calculations. Awesome!

• Ur mum gay

7 months ago

This is a decent website

• Christopher

8 months ago

Wow this is really helpful thanks

• Michael

10 months ago

Hi,

I'm wrapping my head around this problem: I know one side, and the two angles produced by the median on the opposing corner. I'd like to know the length of the other two sides. I drew a scheme, available here:

www.Stavrox.com/image/Triangle.png

The green values are known (a, alpha, beta) , I'd like to calculate b, c and also x. Can you help me.

• Ferny Vise

10 months ago from San Francisco, CA

I really like this article. As a math major myself, I believe math is beautiful!

• Oscar Skabar

12 months ago

I have an example I cannot work out..... Two birds sitting on a 90 degree mask one at 9m up & the other at 6m up but are 15m apart from each other, they see a fish in the water, how do I calculate the distance of the fish from the birds so they are equal in distance

• Rodrigo

12 months ago

Hi, Eugene! You can calc the three angles inside a triangle using tangent half-angle like this:

tan(alpha/2) = r / (p-a)

tan(beta/2) = r / (p-b)

tan(gamma/2) = r / (p-c)

p = (a+b+c) / 2 (semiperimeter)

r = sqrt( (p-a)(p-b)(p-c) / p )

alpha + beta + gamma = 180 (they are the internal angles of the triangle :)

• AUTHOR

Eugene Brennan

13 months ago from Ireland

Hi Carla. There may be a simpler way of doing it, but you can use the cosine rule in reverse to work out the angle B. Then since it's bisected, you know half this angle. Then use the cosine rule in reverse or the sine rule to work out the angle between sides AB and CA. You know the third angle (between the bisector line and side CA) because the sum of angles is 180 degrees. Finally use the sine rule again to work out the distance from A to the bisection point knowing the length of AB and half the bisected angle.

• Carla

13 months ago

Triangle ABC have sides AB=42 BC 64 and CA 84. At what distance from A along AC will the other end bisector of angle B located

• AUTHOR

Eugene Brennan

13 months ago from Ireland

You can't find side lengths with angles alone. Similar triangles have the same angles, but the sides are different. You must have the length of at least one side and two angles.

• william

13 months ago

how do you find side lengths with only angle measurements

• AUTHOR

Eugene Brennan

13 months ago from Ireland

Hi Faria, If you don't know any of the angles, you need to know the lengths of all the sides.

• faria

13 months ago

how to find the measurement if none of the angle of triangle is given??

• AUTHOR

Eugene Brennan

13 months ago from Ireland

If you have the angle at each end, then you can work out the third angle because you know all the angles add up to 180 degrees. Then use the sine rule to work out each side (see example above in the text)

• Karen

13 months ago

i have the the length of one side and the angle at each end, what is the sum to work out the length of the other sides

• AUTHOR

Eugene Brennan

13 months ago from Ireland

Hi Tom,

If you know the lengths of all three sides, use the cosine rule first and the arccos function to work out one of the angles. Then use the sine rule (or the cosine rule again) to work out the one of the other two angles and the fact that they add up to 180 degrees to find the last angle

As regards Excel, I've added a photo to the article showing how to implement a formula for working out an angle using the cosine rule.

• tom sparks

13 months ago

I have a right angled triangle and know the lengths of all three sides. I would like to calculate the other angles.

I have tried TAN in Excel but it says using this 'Returns the tangent of the given angle,.

What would be the best way to work this out

Hope you can help

Kind regards

• AUTHOR

Eugene Brennan

13 months ago from Ireland

• Sanjeev

13 months ago

Right angle and h is 421.410

How find 2 angles and two sides.

• AUTHOR

Eugene Brennan

14 months ago from Ireland

You kneed to know at least one other angle or length. The exception is a right-angled triangle. If you know one angle other than the right angle, then you can work out the remaining angles using sine and cos relationships between sides and angles and Pythagoras' Theorem.

• SUDHAKAR G

14 months ago

how to i find the length in a Scalene triangle? we konw only one angle and one length.

• AUTHOR

Eugene Brennan

15 months ago from Ireland

If two sides are given and the angle between them, use the cosine rule to find the remaining side, then the sine rule to find the other side.

If the angle isn't between the known side, use the sine rule to find the angles first, then the unknown side.

You at least need to know the angle between the sides or one of the other angles so in your example it's the sine rule you need to use.

• Akhyar

15 months ago

If only two sides are given of a non right angled triangle .. then how to find angle between them

• AUTHOR

Eugene Brennan

17 months ago from Ireland

Hi Imran,

There's an infinite number of solutions for angles A and B and sides a and B. Draw it out on a piece of paper and you'll see that you can orientate side c with a known length (e.g. pick a length of 10 cm) and change the angles A and B to what ever you want.

You need to know either the length of one more side or one more angle.

• Imran Hussain from India

17 months ago

Call the angles A,B and C and the lengths of the sides a, b and c.

a is opposite A

b is opposite B

c is opposite C

C is the right angle = 90º and c is the hypotenuse.

How to find the sides of triangle a and b and other 2 angles A and B, if i know only angle C and side c which is hypotenuse?

• AUTHOR

Eugene Brennan

18 months ago from Ireland

Hi Liam,

You need to know at least one of the sides.

You could have a very large or very small triangle with the same angles. These are called similar triangles. See the diagram in the tutorial.

• Liam

18 months ago

How do I find a side in a right angle triangle if I know all three angles but no sides?

• AUTHOR

Eugene Brennan

18 months ago from Ireland

If the holes are equally spaced around the imaginary circle, then the formula for the radius of the circle is:

R = B / (2Sin(360/2N))

B is the distance between holes

N is the number of holes

• Divya

18 months ago

how to calculate distance of each hole at PCD from centre circle

• Amar36

20 months ago

Hi sir

how is that possible to know angle by just having ratios of two heights of triangle and u need not use protector or some other instruments and not even inverse trigonometric functions just simply by ratio do we calculate them or not if then how

I asked it because how they have founded the angles of different triangles with it any discovery of inverse trigonometric functions.

• AUTHOR

Eugene Brennan

21 months ago from Ireland

You're welcome Johanese!

• Johanese Tommy

21 months ago

THANKS VERY MUCH FOR THIS LESSONS I REALLY ENJOY IT

• AUTHOR

Eugene Brennan

22 months ago from Ireland

No enough information shahid! If you think about it, there's an infinite number of triangles that satisfy those conditions. Area = (1/2) base x height. So there's no unique values of base and height to satisfy equation (1/2) base x height = 10 m squared.

• shahid abbasi

22 months ago

area of right angle triangle is 10m and one angle is 90degree then how calculate three sides and another two angles.

• AUTHOR

Eugene Brennan

23 months ago from Ireland

If you assign lengths to all sides, you easily can work out the angles. Which sides did assign a length to?

• Gem

23 months ago

Any luck Eugene? I have figured out some of the angles by folding a part of the paper that can let me use trig to figure it out if I assign each side a length.

• AUTHOR

Eugene Brennan

23 months ago from Ireland

Hi Danya,

Because you know two of the angles, the third angle can simply be worked out by subtracting the sum of the two known angles from 180 degrees. Then use the Sine Rule described above to work out the two unknown sides.

• danya61

23 months ago

Hi

I have a triangle with two known angles and one known length of the side between them, and there is no right angle in the triangle. I want to calculate each of unknown sides. How can I do that? (The angle between unknown sides is unknown.)

• AUTHOR

Eugene Brennan

23 months ago from Ireland

Draw a diagram jeevan. I can't really visualize this.

• jeevan

23 months ago

there are 3 circles 1 large circle is a pitch circle having 67 diameter and medium circle is drawn on the circumference of pitch circle at the angle of 5 degree hvaing 11.04 radius and a small circle with only moves in x y direction on pitch circle radius having 1.5 radius so if the medium circle is moved 5degree then at which point the small circle is coinciding and the distance from small circle to center of large/pitch circle.?

• Gem

23 months ago

It is tough to prove for sure. I thought I had it by assigning each side a random length ( such as 2cm) and then taking the middle point as half, which looked like the right angle triangle on the top right hand side was half of the half. But it still can't be proven to be half because of the fold.

• AUTHOR

Eugene Brennan

24 months ago from Ireland

If it's an equilateral triangle, the sides and angles can be easily worked out. Otherwise the triangle can have an infinite number of possible side lengths as the apexes A and C are moved around. So if none of the magnitudes of lengths are known, the expression for lengths of sides of the triangle and its angles would have to be expressed in terms of the square's sides and the lengths AR and CP?

• Gem

24 months ago

The whole problem has no measurements or angles. It only has angle names such as A,B,C,D etc. My starting point is from the common knowledge that a square has 4 x 90 degree angles. If I could determine one other angle then I could figure out the whole problem by using the 180 degree rule of triangles. I will snap a picture of it and try and upload it here on Monday, or sketch and upload it. It seems to be a real stumper, 2/70 people at a workshop were able to figure it out, as I was told by the person who passed it along to me. I appreciate your reply, and I look forward to sharing the appropriate visual information with you.

• AUTHOR

Eugene Brennan

24 months ago from Ireland

Hi Gem,

Is any information given about where the corners of the triangle touch the sides of the square or the lengths of the square's sides? If the triangle isn't equilateral (or even if it is), it seems that there would be an infinite number of placing the triangle in the square.

• Gem

24 months ago

Problem: A triangle is placed inside a square. The triangle doesn't have measurements or any listed angles. So we can't identify the type (although it looks equilateral) or make any concrete assumptions about the triangle. I'm suppose to figure out the angles of the triangle without a protractor or ruler based on the only angles I am given which are the 90 degrees from each corner of the square it's in. Since the lines that cut through the square from the main triangle inside the square make new sets of smaller triangles, I still can't make out complimentary or supplementary angles since most of those smaller triangles aren't definitely right angles isosceles triangles.

I'm not sure if my question is clear, so if you answer back I'll try and add a picture or sketch to clarify.

Just picture a square with a triangle in it touching all 3 sides of its points to the square with no units of measure and no angles. We can only assume that the square has 90 degree angles in the corners and that's all we are given to work with.

Thanks Gem

• AUTHOR

Eugene Brennan

2 years ago from Ireland

Hi Maxy,

Call the angles A,B and C and the lengths of the sides a, b and c.

a is opposite A

b is opposite B

c is opposite C

C is the right angle = 90º and c is the hypotenuse.

If the angle A is known and the side opposite it, a, is known

Then Sin A = opposite/hypotenuse = a/c

So c = a/Sin A

Since you know a and A, you can work out c.

Then use Pythagoras's theorem to work out b

c² = a² + b²

So b² = c² - a²

So b = √(c² - a²)

If the angle A is known and the side adjacent to it, b, is known

Then Cos A = adjacent/hypotenuse = b/c

So c = b / Cos A

Since you know b and A, you can work out c.

Then use Pythagoras's theorem to work out a.

c² = a² + b²

So a² = c² - b²

So a = √(c² - b²)

• Maxy

2 years ago

How to calculate hypoyeneous and side of right angled triangle, if length of one side is given.

• AUTHOR

Eugene Brennan

2 years ago from Ireland

You need to use the cosine rule in reverse.

So if the angles are A, B, and C and the sides are a,b and c.

Then c² = a² + b² - 2abCos C

Rearranging gives angle C = Arccos ((a² + b² - c²) / 2ab)

You can work out the other angles similarly using the cosine rule. Alternatively use the sine rule:

So a/Sin A = c/Sin C

So Sin A = a/c (Sin C)

and A = Arccos ( a/c (Sin C) )

and similarly for the other angles

• Hannah

2 years ago

How do you find the angle if all three sides are given

• AUTHOR

Eugene Brennan

2 years ago from Ireland

Polygons are a lot more complicated than triangles because they can have any number of sides (they do of course include triangles and squares). Also polygons can be regular (have sides the same length) or non-regular (have different length sides).

Here's two formulae:

For a regular or non-regular polygon with n sides

Sum of angles = (n-2) x 180 degrees

For a regular convex polygon (not like a star)

Interior angles = (1 - 2/n) x 180 degrees

• AUTHOR

Eugene Brennan

2 years ago from Ireland

Hi Jeetendra,

This is called a scalene triangle. The longest edge of any triangle is opposite the largest angle. If all angles are known, the length of at least one of the sides must be known in order to find the length of the longest edge. Since you know the length of an edge, and the angle opposite it, you can use the sine rule to work out the longest edge. So if for example you know length a and angle A, then you can work out a/Sin A.

If c is the longest side,

then a/sin A = c/Sin C ,

so rearranging,

c = a Sin C / Sin A

a, C and A are known, so you can work out c

• Jeetendra Beniwal( from India)

2 years ago

If all three angles are given then how we find largest edge of triangle,if all angles are acute

• AUTHOR

Eugene Brennan

3 years ago from Ireland

Thanks Ron, triangles are great, they crop up everywhere in structures, machines, and the ligaments of the human body can be thought of as ties, forming one side of a triangle.

• Ron Bergeron

3 years ago from Massachusetts, US

I've always found the math behind triangles to be interesting. I'm glad that you ended the hub with some examples of triangles in every day use. Showing a practical use for the information presented makes it more interesting and demonstrates a purpose for learning about it.

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