How to Calculate the Sides and Angles of Triangles
Basics of Triangles
In this tutorial, you'll learn the basic facts about triangles, 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.
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What Is a Triangle?
By definition, a triangle is a polygon with three sides.
Polygons are plane (flat, twodimensional) 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.
Basic Facts About Triangles
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. (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.
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 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
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.

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.

Triangle Types and Classifications
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 = 4c = 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:
c^{2} = a^{2} + b^{2}  2abCos C
The Cosine Rule Should Be Used If ...
 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.
 You know all the lengths of the sides but none of the angles.
Then, by rearranging the cosine rule equation:
C = Arccos ((a^{2 }+ b^{2}  c^{2}) / 2ab)
The other angles can be worked out similarly.
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, Tsquare, 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 fullsizeHow 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.
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

FAQs About Triangles
Below are some frequently asked questions about triangles.
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 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?
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:
a^{2} = b^{2} + c^{2}  2bc cos A
or
b^{2} = a^{2} + c^{2}  2ac cos B
or
c^{2} = a^{2} + b^{2}  2ab cos C
How Do I Calculate the Volume of a Triangle?
Since a triangle is a plane and twodimensional object, it is impossible to discover its volume. A triangle is flat. Thus, it has no volume.
Triangular prisms, on the other hand, are threedimensional 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:
c^{2} = a^{2} + b^{2}  2abCos C
Then, by rearranging the cosine rule equation, you can work out the angle
C = Arccos ((a^{2 }+ b^{2}  c^{2}) / 2ab)
and
B = Arccos ((a^{2 }+ c^{2}  b^{2}) / 2ac)
The third angle A is (180  C  B)
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.
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.
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
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.
Helpful 91How 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 rightangled triangle.
Helpful 39What 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.
Helpful 39How 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
Helpful 27How should I approach the problem  The triangles ABC and ACD are such that BC 32 cm, AD  19cm, CD  28cm BAC  74 ( angle ) and ADC  67 ( angle )?
Use the cosine rule to work out AC. Then the sine rule to work out the remaining angles/sides.
Helpful 4
© 2016 Eugene Brennan
Comments
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,
thanks
Wow, amazing! One of the best works I ever have seen here!
A good review of many years of wining and dining with math calculations. Awesome!
This is a decent website
Wow this is really helpful thanks
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.
I really like this article. As a math major myself, I believe math is beautiful!
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
Hi, Eugene! You can calc the three angles inside a triangle using tangent halfangle like this:
tan(alpha/2) = r / (pa)
tan(beta/2) = r / (pb)
tan(gamma/2) = r / (pc)
p = (a+b+c) / 2 (semiperimeter)
r = sqrt( (pa)(pb)(pc) / p )
alpha + beta + gamma = 180 (they are the internal angles of the triangle :)
Congrats for your site!
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
how do you find side lengths with only angle measurements
how to find the measurement if none of the angle of triangle is given??
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
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
Right angle and h is 421.410
How find 2 angles and two sides.
how to i find the length in a Scalene triangle? we konw only one angle and one length.
If only two sides are given of a non right angled triangle .. then how to find angle between them
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?
How do I find a side in a right angle triangle if I know all three angles but no sides?
how to calculate distance of each hole at PCD from centre circle
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
THANKS VERY MUCH FOR THIS LESSONS I REALLY ENJOY IT
area of right angle triangle is 10m and one angle is 90degree then how calculate three sides and another two angles.
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.
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.)
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.
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.
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.
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
How to calculate hypoyeneous and side of right angled triangle, if length of one side is given.
How do you find the angle if all three sides are given
If all three angles are given then how we find largest edge of triangle,if all angles are acute
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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