How to Calculate the Sides and Angles of Triangles
The Triangle—a 3Sided 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. 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.
All the Angles Add Up To 180 degrees
For every triangle in the known Universe, all the angles add up to 180 degrees.
Similar Triangles
Similar triangles have exactly the same angles but different length sides.
The sides are in the same ratio.
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
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 (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 rightangled 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 = 4So according to Pythagoras:
c² = a² + b²
So c² = 3² + 4² = 9 + 16 = 25
and c = √25 = 5
Sine and Cosine
A rightangled 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 rightangled 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.
sine θ = length of opposite side / length of hypotenuse
cosine θ = length of adjacent side / length of hypotenuse
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
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.
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:
c^{2} = a^{2} + b^{2}  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 ((a^{2 }+ b^{2}  c^{2}) / 2ab)
The other angles can be worked out similarly.
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, Tsquare 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.
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

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 newfangled 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.
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 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.
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:
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
© 2016 Eugene Brennan
Comments
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
Good explanation. Thank
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.
In the triangle below, what is the value of ( a ْ + b ْ ) ?
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.
It's kind of like the one in the link below. But remember that we have no tools such as rulers to measure the shapes. We have to figure out what all those letters represent or are equal to in angles. Don't attribute the triangle as being Scalene, Equilateral or any other by eye, just try and figure out from the 90 degree angles how we can solve it.
Good luck ha ha
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
Tell me steps to solve easily polygons with sides
If all three angles are given then how we find largest edge of triangle,if all angles are acute
Please explain more example......in every pattern
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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