How to Calculate the Sides and Angles of Triangles
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.
If you find the article useful, please share it on Facebook or Pinterest.
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.
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 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 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
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:
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.
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.
Summary - Which Rules Do I Use?
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.
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.
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
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
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