How to Calculate the Sides and Angles of Triangles

Updated on June 14, 2018
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Eugene is a qualified control/instrumentation engineer Bsc (Eng) and has worked as a developer of electronics & software for SCADA systems.

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The Triangle—a 3-Sided Polygon

Polygons are plane (flat) shapes with several straight sides. Examples are squares, triangles, hexagons, and pentagons. 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 3 sides.

In this tutorial, you'll learn about Pythagoras's 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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How to Work Out the Sides and Angles of Triangles?

We can use:

  • Pythagoras's Theorem
  • Sine Rule
  • Cosine Rule
  • The fact that all angles add up to 180 degrees

Triangles Have Angles

Triangles have 3 corners and the angle between the sides can be anything from greater than 0 up to 180 degrees, half the angle in a full circle which is 360 degrees. The angles can't be 0 or 180 degrees because then the triangles would become straight lines.

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

Angles of a triangle range from  0 to   180 degrees
Angles of a triangle range from 0 to 180 degrees | Source
Triangles classed by side and angles
Triangles classed by side and angles | Source

All the Angles Add Up To 180 degrees

For every triangle in the known Universe, all the angles add up to 180 degrees.

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

Similar Triangles

Similar triangles have exactly the same angles but different length sides.
The sides are in the same ratio.

Similar triangles
Similar triangles

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 seen the character μ (mu) to 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

Using Trigonometry For Solving Triangles

Trigonometry is a branch of mathematics which deals with the relationship between the lengths of the sides of a triangle and its angles. If we don't know all the lengths of the sides of a triangle and/or don't know all the angles, there are several equations and rules that can be used to work them out. Working out sides and angles is known as solving triangles. Below we'll use the Sine Rule, Cosine Rule, Pythagoras's Theorem and the fact that all angles sum to 180 degrees to solve triangles.

Pythagoras's Theorem
Pythagoras's Theorem | Source

Pythagoras's Theorem (The Pythagorean Theorem)

Pythagoras's Theorem (also known as the Pythagorean theorem) states that for a right angled triangle

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

The hypotesuse is the longest side of a right-angled triangle, opposite the right angle

So if you know the lengths of 2 sides, all you have to do is square the two lengths, add the results and then take the square root of the sum to get the length of the hypotenuse.

Example: "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

So according to Pythagoras:

c² = a² + b²

So c² = 3² + 4² = 9 + 16 = 25

and c = √25 = 5

Sine, Cosine and Tan of an Angle

A right-angled triangle has one angle measuring 90 degrees. The side opposite this angle is known as the hypotenuse and it is the longest side. Sine and cosine are trigonometric functions of an angle and are the ratios of the lengths of the other two sides to the hypotenuse of a right-angled triangle.

In the diagram below, one of the angles is represented by the Greek letter θ.

The side a is known as the "opposite" side and side b is the "adjacent" side 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, not necessarily an angle in a triangle, so it's possible to just have two lines meeting at a point and to evaluate sine or cos for that angle. However sine and cos are derived from the sides of an imaginary right angled 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 and adjacent sides and hypotenuse can be determined.

Over the 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.

Sine and cosine are sometimes abbreviated to sin and cos

Sine, cosine and tan
Sine, cosine and tan

The Sine Rule

The ratio of the length of a side of a triangle to the sine of the angle opposite it is constant for all 3 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 can be used:

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

Sine rule
Sine rule | Source
Sine rule example
Sine rule example | Source

The Cosine Rule

For a triangle with sides a, b and c, then if a and b are known and C is the included angle (the angle between the sides), c can be worked out from:

c2 = a2 + b2 - 2abCos C

The Cosine Rule can be used if:

(a) 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.

(b) You know all the lengths of the sides but none of the angles.

Then rearranging the cosine rule equation:

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

The other angles can be worked out similarly.


Cosine rule
Cosine rule | Source
Cosine rule example
Cosine rule example | Source

Area of a Triangle

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. On a drawing, 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 formula 1/2ah to get the area.

Alternative Method of Working Out Area

The simple method above requires that you actually measure the height of a triangle. If you know the length of 2 of the sides and the included angle, you can work out the area analytically.

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

Summary - Which Rules Do I Use?

Known Parameters
Rule to Use
Triangle is right handed and I know length of 2 sides
Use Pythagoras's Theorem to work out remaining side and Sine Rule to work out angles
Triangle is right handed and I know the length of 1 side and one angle
Use the trigonometric identities sine and cos to work out other sides and sum of angles = 180 to work out remaining angle
I know the length of 2 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 1 side and all 3 angles
Use the Sine Rule to work out remaining sides
I know the lengths of all 3 sides
Use the Cosine Rule in reverse to work out each angle. C = Arccos ((a² + b² - c²) / 2ab)

How to Measure Angles

You can use a protractor, or alternatively a more new-fangled 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.

An unknown angle
An unknown angle | Source
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
Source
Tacklife MDA01 Digital Protractor 7 Inch Angle Finder Stainless Steel Ruler  With HOLD and Zeroing Function, Large LCD Display, Coin Battery Included for Woodworking, Construction, Repairing
Tacklife MDA01 Digital Protractor 7 Inch Angle Finder Stainless Steel Ruler  With HOLD and Zeroing Function, Large LCD Display, Coin Battery Included for Woodworking, Construction, Repairing

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.

 

Triangles Are Strong!

A triangle is the most basic polygon and can't be pushed out of shape easily unlike a square. If you look closely enough, triangles are used everywhere in machines and structures because the shape is so strong.
One of the strengths of a triangle is due to the fact that when any of the corners are loaded, the side opposite acts as a tie, undergoing tension and preventing a framework from deforming. So for example on a roof truss, the horizontal ties provide strength and prevent the roof spreading out at the eaves. The sides of a triangle can also act as struts, undergoing compression. An example is a shelf bracket or struts on the underside of an aeroplane wing or tail.

Truss bridge
Truss bridge | Source
A roof truss
A roof truss | Source
Source
Source
Spoked wheels
Spoked wheels | Source
Can you spot the triangles?
Can you spot the triangles? | Source
Struts on the underside of an aeroplane tail.
Struts on the underside of an aeroplane tail. | Source

Using a Triangle Calculator

There are lots of online triangle calculators on the web which make things easy (or take the fun out of doing the calculations!). These calculators allow you to enter angles and side lengths and they will work out all the remaining sides and angles. Here's one you can try:

http://www.trianglecalculator.net

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

  • Which formula is used when given 90-degree triangle, opposite angle is 26 degrees and one leg is know?

    Use the fact that the cos of an angle is the length of the adjacent side divided by the hypotenuse, or the sine of an angle is the opposite side divided by the hypotenuse. In your case, you know the side opposite the angle.

    So sine (26 degrees) = length opposite side / length hypotenuse

    Therefore

    Length hypotenuse = length opposite side / sine (26 degrees)

    Use Pythagoras's theorem to work out remaining side

    and remaining angle = 180 - (90 + 26) = 64 degrees

  • How do I know when to use sine or cosine formula when given two degrees and one length?

    If the length is opposite one of the known angles, you can use the Sine Rule. If it isn't, you can work out the third angle since the three angles sum to 180 degrees. Then use the Sine Rule. The Cosine Rule is normally used when you only have one angle between two known sides.

  • How do I know when to use the sine or cosine formula?

    If you know the length of two sides and the angle between them, then you can use the cosine formula to work out the remaining side. Otherwise, the sine formula or Pythagorean theorem can be used.

  • What rule would be used to find the length of sides if all three angles are known?

    There is an infinite number of similar triangles that have the same angles. Imagine if you have a triangle and you know all the angles. You can keep making it bigger, but the angles stay the same. However, the sides get longer. So you need to know the length of at least one side. Then you can use the Sine Rule to work out the remaining three sides.

  • What is the maximum and minimum value for the sine of an angle?

    If θ is the angle, the maximum value of sine occurs when θ = 90 degrees or π/2 radians. The minimum value is -1 and this occurs when θ = 270 degrees or 3π/2 radians.

© 2016 Eugene Brennan

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    • eugbug profile imageAUTHOR

      Eugene Brennan 

      7 weeks 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 

      7 weeks 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 

      7 weeks 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 

      7 weeks ago

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

    • profile image

      Amar36 

      3 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 

      3 months ago

      Good explanation. Thank

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      3 months ago from Ireland

      You're welcome Johanese!

    • profile image

      Johanese Tommy 

      3 months ago

      THANKS VERY MUCH FOR THIS LESSONS I REALLY ENJOY IT

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      5 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 

      5 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 

      6 months ago

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

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      6 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 

      6 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 

      6 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 

      6 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 

      6 months ago from Ireland

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

    • profile image

      jeevan 

      6 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 

      6 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 

      6 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 

      7 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 

      7 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 

      7 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 

      7 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 

      7 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 

      7 months ago

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

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      7 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 

      7 months ago

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

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      7 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 

      7 months ago

      Tell me steps to solve easily polygons with sides

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      7 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) 

      7 months ago

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

    • profile image

      katiwal 

      8 months ago

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

    • eugbug profile imageAUTHOR

      Eugene Brennan 

      24 months 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 

      24 months 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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