How to Calculate the Sides and Angles of Triangles Using Pythagoras' Theorem, Sine and Cosine Rule

In this tutorial, we'll first learn the absolute basics about triangles. Then we'll learn about trigonometry which is a branch of mathematics that covers the relationship between the sides and angles of triangles.

What's Covered in the Tutorial?

• Polygons and the Definition of a Triangle
• The Basic Facts About Triangles
• The Triangle Inequality Theorem
• Different Types of Triangles
• Using the Greek Alphabet for Equations
• Sine, Cosine and Tangent
• Pythagoras's Theorem
• The Sine and Cosine Rules
• How to Work Out the Sides and Angles of a Triangle
• Measuring Angles
• How to Calculate the Area of a Triangle

 

What Is a Triangle?

By definition, a triangle is a polygon with three sides.

Polygons are plane shapes with several straight sides. "Plane" just means they're flat and two-dimensional. Other examples of polygons include squares, pentagons, hexagons and octagons. The word plane originates from the Greek polús meaning "many" and gōnía meaning "corner" or "angle." So polygon means "many corners." A triangle is the simplest possible polygon, having only three sides.

The Symbol for Degrees

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

Angles of a triangle range from greater than 0 to less than 180 degrees. © Eugene Brennan

Basic Facts About Triangles

• A triangle is a polygon with three sides.
• All the internal angles add up to a total of 180 degrees.
• The angle between two sides can be anything from greater than 0 to less than 180 degrees.
• The angle between two sides can't be 0 or 180 degrees, because the triangle would then become three straight lines superimposed on each other (These are called degenerate triangles).
• Similar triangles have the same angles, but different length sides.

 

No matter what the shape or size of a triangle, the sum of the 3 internal angles is 180. © Eugene Brennan

Similar triangles have the same angles but different length sides. © Eugene Brennan

 

What Are the Different Types of Triangles?

Before we learn how to work out the sides and angles of a triangle, it's important to know the names of the 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

 

Type of Triangle by Lengths of Sides	Description
Isosceles	An isosceles triangle has two sides of equal length, and one side that is either longer or shorter than the equal sides.
Equilateral	All sides and angles are equal in length and degree.
Scalene	All sides and angles are of different lengths and degrees.

Types of triangles by length of sides.

 

Type of Triangle by Internal Angle	Description
Right (right angled)	One angle is 90 degrees.
Acute	Each of the three angles measures less than 90 degrees.
Obtuse	One angle is greater than 90 degrees.

 Types of triangles by angle.

Triangles classified by side and angles. © Eugene Brennan

What is Trigonometry?

Trigonometry is branch of mathematics that covers the relationship between the sides and angles of triangles. If we don't know all the sides or angles, we can use trigonometry to work out the unknown quantities.

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

There are several methods that can be used to calculate the unknown sides or angles. We can use formulas, mathematical rules, or the fact that the angles of all triangles add up to 180 degrees.

Tools to discover the sides and angles of a triangle

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

Pythagoras's Theorem (The Pythagorean Theorem)

Pythagoras's theorem uses trigonometry to calculate 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.

Pythagoras's Theorem. The hypotenuse is the longest side, opposite the right angle that measures 90 degrees. © Eugene Brennan

In the diagram above, the hypotenuse is the longest side of the right triangle, and is located opposite the right angle which is the angle that measures 90 degrees. 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.

If the sides of a triangle are a, b and c and c is the hypotenuse, Pythagoras's Theorem states that:

c² = a² + b² 

So

c = √(a² + b²)

 

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

So c² = 25 and to find c, we just take the square root of 25 giving:

c = √25= 5

What are the Sine, Cosine and Tangent 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 of this type of triangle). The length of the hypotenuse can be discovered using Pythagoras's 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. (pronounced "thee - taa" ).

Sine, cosine and tan. © Eugene Brennan

 

Side a is known as the "opposite" side.
Side b is called the "adjacent" side.

The vertical lines "||" around the words below mean "length of."

So sine, cosine and tangent are defined as follows:

sine θ = |opposite side| / |hypotenuse| 

cosine θ = |adjacent side| / |hypotenuse|

tan θ = |opposite side| / |adjacent side|

Sine and cosine apply to an angle, not just an angle in a triangle, so it's possible to have two lines meeting at a point and to evaluate sine or cosine for that angle even though there's no triangle as such. However, sine and cosine are derived from the sides of an imaginary right triangle superimposed on the lines.
For instance, in the second diagram above, the purple triangle is scalene not right angled. However, you can imagine a right-angled triangle superimposed on the purple triangle, from which the opposite, adjacent and hypotenuse sides can be determined.
Over a range 0 to 90 degrees, sine ranges from 0 to 1, and cosine 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 above 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, cosine and tangent are often abbreviated to sin, cos and tan respectively.

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

The sine rule. © Eugene Brennan

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

When should the sine rule be used?

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.

Example showing how to use the sine rule to calculate the unknown side c. © Eugene Brennan

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 = a² + b² - 2ab cos C

 

The cosine rule. © Eugene Brennan

When should the cosine rule be used?

1. You know the lengths of the 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 the lengths of all the sides but none of the angles. Rearranging the cosine rule equation gives the length of one of the sides.

c = a² + b² - 2ab cos C

Rearranging the equation:

C = arccos ((a² + b² - c²) / 2ab)

The other angles can be worked out similarly.

Example using the cosine rule. © Eugene Brennan

How to Find the Angles of a Triangle Knowing the Ratio of the Side Lengths

If you know the ratio of the side lengths, you can use the cosine rule to work out two angles then the remaining angle can be found knowing all angles add to 180 degrees.

Example:

A triangle has sides in the ratio 5:7:8. Find the angles.

Answer:

So call the sides a, b and c and the angles A, B and C and assume the sides are a = 5 units, b = 7 units and c = 8 units. It doesn't matter what the actual lengths of the sides are because all similar triangles have the same angles. So if we work out the values of the angles for a triangle which has a side a = 5 units, it gives us the result for all these similar triangles.

Use the cosine rule. So c² = a² + b² - 2ab cos C

Substitute for a,b and c giving:

8² = 5² + 7² - 2(5)(7) cos C

Working this out gives:

64 = 25 + 49 - 70 cos C

Simplifying and rearranging:

cos C = 1/7 and C = arccos(1/7).

You can use the cosine rule again or sine rule to find a second angle and the third angle can be found knowing all the angles add to 180 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 remaining angle can be calculated. Use the sine rule to work out the two unknown sides
I know the length of a side and one angle		You need to know more information, either one other side or one other angle. The exception is if the known angle is in a right angled triangle and not the right angle.

A summary of how to work out angles and sides of a triangle.

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. Using the perpendicular height

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.

Working out the area of a triangle from the base length and perpendicular height. © Eugene Brennan

Method 2. Using side lengths and angles

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).

Working out the area of a triangle from the lengths of two sides and the sine of the included angle. © Eugene Brennan

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

Using Heron's formula to work out the area of a triangle. © Eugene Brennan

Using the Greek Alphabet for Equations

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.
You may have 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. λ (lambda) is used for wavelength, and, of course, π (pi) is the ratio of the circumference to the diameter of a circle.
In trigonometry, the characters θ (theta), φ (phi) and some others are often used for representing angles.

Letters of the Greek alphabet. © Eugene Brennan

How Do You Measure Angles?

Digital finder. Image courtesy Amazon

You can use a protractor or a digital angle finder like the one above from Amazon.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 an angle finder as a replacement for a bevel gauge for transferring angles e.g. when marking the ends of rafters before cutting. Accuracy is usually down to 0.1 degrees.

You can draw and measure angles with a protractor. © Eugene Brennan

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 (which can be joists in a ceiling) 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 a light aircraft wing or the tail wing itself.

Truss bridge. Image courtesy Kanenori on Pixabay

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.

Calculating side lengths in Excel using the cosine rule. © Eugene Brennan

FAQs About Triangles

Below are some frequently asked questions about triangles.

What do the angles of a triangle add up to?

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 depends 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 right angled triangle?

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:

a² = b² + c² - 2bc cos A

or

b² = a² + c² - 2ac cos B

or

c² = a² + b² - 2ab cos C

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:

c² = a² + b² - 2ab cos C

Then, by rearranging the cosine rule equation, you can work out the angle

C = arccos ((a² + b² - c²) / 2ab)and

B

= arccos ((a²+ c² - b²) / 2ac)

The third angle A is (180 - C - B)

How to find the perimeter of a triangle

Finding the perimeter of a triangle is a straightforward operation. The perimeter is equivalent to the added lengths of all three sides.

perimeter = a + b + c

How to find the height of a triangle

Finding the height of a triangle is easy if you have the triangle's area. If you're given the area of the triangle:

height = 2 x area / base

If you don't have the area, but only have the side lengths of the triangle, use the following:

height = 0.5 x √ ((a + b + c)(-a + b + c)(a - b + c)(a + b - c)) / b

If you only have two sides and the angle between them, try this formula:

area = 0.5 (a)(b)(sin(γ)), then

height = area(sin(γ))

References

1. Trigonometry. from Wolfram MathWorld. (n.d.). Retrieved May 24, 2022, from https://mathworld.wolfram.com/topics/Trigonometry.html

2. Equilateral triangle. from Wolfram MathWorld. (n.d.). Retrieved May 24, 2022, from https://mathworld.wolfram.com/EquilateralTriangle.html

3. Isosceles Triangle. from Wolfram MathWorld. (n.d.). Retrieved May 24, 2022, from https://mathworld.wolfram.com/IsoscelesTriangle.html

4. Scalene Triangle. from Wolfram MathWorld. (n.d.). Retrieved May 24, 2022, from https://mathworld.wolfram.com/ScaleneTriangle.html

5. Prof. David E. Joyce. The laws of cosines and Sines. Laws of Cosines & Sines. (n.d.). Retrieved May 24, 2022, from https://www2.clarku.edu/faculty/djoyce/trig/laws.html

Disclaimer

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualised advice from a qualified professional.

© 2016 Eugene Brennan