How to Calculate the Sides and Angles of Triangles

Updated on September 8, 2018
eugbug profile image

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

Source

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.

In this tutorial, you'll 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.

If you find the article useful, please share it on Facebook or Pinterest.

What Are the Different Types of Triangles?

Before we learn how to discover the sides and angles of a triangle, it is important to know the many 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

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.
Right
One angle is 90 degrees.
Acute
Each of the three angles measure less the 90 degrees.
Obtuse
One angle is greater than 90 degrees.
Triangles classed by side and angles.
Triangles classed by side and angles. | Source

Using the Greek Alphabet for Equations

Another topic that must be covered 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.

The Greek alphabet.
The Greek alphabet. | Source

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

There are many methods available to the mathematician 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

Before we delve into Pythagoras' theorem, the sine rule, and the cosine rule, it is important to state that all triangles have three corners with angles that 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.

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.

Click thumbnail to view full-size
No matter what the shape or size of a triangle, the sum of the 3 angles is 180 degrees.Angles of a triangle range from 0 to 180 degrees.Similar triangles.
No matter what the shape or size of a triangle, the sum of the 3 angles is 180 degrees.
No matter what the shape or size of a triangle, the sum of the 3 angles is 180 degrees. | Source
Angles of a triangle range from 0 to 180 degrees.
Angles of a triangle range from 0 to 180 degrees. | Source
Similar triangles.
Similar triangles. | Source

Pythagoras' Theorem (The Pythagorean Theorem)

Pythagoras' theorem uses trigonometry to discover the longest side (hypotenuse) of a right triangle. 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

Pythagoras's Theorem
Pythagoras's Theorem | Source

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
Sine, cosine, and tan.Sine rule.Sine rule example.Cosine rule.Cosine rule example.
Sine, cosine, and tan.
Sine, cosine, and tan. | Source
Sine rule.
Sine rule. | Source
Sine rule example.
Sine rule example. | Source
Cosine rule.
Cosine rule. | Source
Cosine rule example.
Cosine rule example. | Source

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

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

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

Which Rule Do I Use?

Known Parameters
Rule to Use
Triangle is right and I know length of two sides.
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
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.
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.
Use the sine rule to work out remaining angles and side.
I know the length of one side and all three angles.
Use the sine rule to work out the remaining sides.
I know the lengths of all three sides
Use the cosine rule in reverse to work out each angle. C = Arccos ((a² + b² - c²) / 2ab)

FAQs About Triangles

Below are some frequently asked questions about triangles.

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, the sum of the sides of a triangle must be calculated after determining the length of each of the sides.

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 Triangle?

Assuming the triangle is right, 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?

There are a few variations of the cosine formula these 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 Many Degrees Are There in a Triangle?

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

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.

Digital Angle Finder Tacklife 7 Inch Digital Protractor 200mm Stainless Steel Ruler, Zeroing/Locking Function, Digital LCD Display, Coin Battery Included for Woodworking, Construction - MDA01
Digital Angle Finder Tacklife 7 Inch Digital Protractor 200mm Stainless Steel Ruler, Zeroing/Locking Function, Digital LCD Display, Coin Battery Included for Woodworking, Construction - MDA01

This digital protractor or angle finder comes in very useful when constructing stuff from wood or metal. I also use it 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.

 
Click thumbnail to view full-size
You can measure an angle with a protractor.You can measure an angle with a digital angle finder.
You can measure an angle with a protractor.
You can measure an angle with a protractor. | Source
You can measure an angle with a digital angle finder.
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
Truss bridge.A roof truss. Cargo loaders.Electric power pylon. Spoked wheels.Airplane support struts.
Truss bridge.
Truss bridge. | Source
A roof truss.
A roof truss. | Source
Cargo loaders.
Cargo loaders. | Source
Electric power pylon.
Electric power pylon. | Source
Spoked wheels.
Spoked wheels. | Source
Airplane support struts.
Airplane support struts. | 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.

Questions & Answers

  • 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 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.

  • 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.

  • 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

© 2016 Eugene Brennan

Comments

    0 of 8192 characters used
    Post Comment

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      3 weeks 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.

    • profile image

      Akhyar 

      3 weeks ago

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

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      2 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.

    • profile image

      Imran Hussain from India 

      2 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?

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      3 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.

    • profile image

      Liam 

      3 months ago

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

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      3 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))

      Where R is the radius

      B is the distance between holes

      N is the number of holes

    • profile image

      Divya 

      3 months ago

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

    • profile image

      Amar36 

      5 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.

      Thank in advance

    • profile image

      Joyce Lamey 

      5 months ago

      Good explanation. Thank

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      6 months ago from Ireland

      You're welcome Johanese!

    • profile image

      Johanese Tommy 

      6 months ago

      THANKS VERY MUCH FOR THIS LESSONS I REALLY ENJOY IT

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      7 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.

    • profile image

      shahid abbasi 

      7 months ago

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

    • profile image

      muntaha 

      8 months ago

      In the triangle below, what is the value of ( a ْ + b ْ ) ?

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      8 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?

    • profile image

      Gem 

      8 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.

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      8 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.

    • profile image

      danya61 

      8 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.)

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      8 months ago from Ireland

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

    • profile image

      jeevan 

      8 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.?

      sir please help me finding the answer thank you.

    • profile image

      Gem 

      8 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.

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      8 months ago from Ireland

      If you draw lines over all the folds, it creates lots of similar triangles, plus what looks like an equilateral triangle although I cant figure out to prove it yet!

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      9 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?

    • profile image

      Gem 

      9 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.

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      9 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.

    • profile image

      Gem 

      9 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

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      9 months 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)

    • profile image

      Maxy 

      9 months ago

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

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      9 months 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

    • profile image

      Hannah 

      9 months ago

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

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      9 months 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

    • profile image

      Fatima 

      9 months ago

      Tell me steps to solve easily polygons with sides

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      9 months 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

    • profile image

      Jeetendra Beniwal( from India) 

      9 months ago

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

    • profile image

      katiwal 

      10 months ago

      Please explain more example......in every pattern

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      2 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.

    • ronbergeron profile image

      Ron Bergeron 

      2 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.

    working

    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, owlcation.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://owlcation.com/privacy-policy#gdpr

    Show Details
    Necessary
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
    Features
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Marketing
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Statistics
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)