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How to solve a triangle given three sides.

 This topic is part of the TCS FREE high school mathematics 'How-to Library'. It shows you how to find the unknown angles of a triangle when given the three side lengths. (See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

### Theory:

This triangle has internal angles 'A', 'B' and 'C', and sides of length 'a', 'b' and 'c':

If three of these six measurements are known, then it may be possible to find the other three.

This is called 'solving' the triangle, and this topic will show you how to solve triangles for the three unknown angles when the three side lengths 'a', 'b' and 'c' are known.

These are the formulas used to solve triangles:

1. The sum of the internal angles equals 180º ...

A + B + C = 180º

2. The 'sine rule' ...

3. The 'cosine rule' ...

a² = b² + c² - 2bc cosA

or

b² = a² + c² - 2ac cosB

or

c² = b² + a² - 2ba cosC

We will now use an example to show how these rules are applied to solve a triangle when the three side lengths are given.

Example: Solve this triangle for the unknown internal angles:

When no angles are known, the cosine rule is the only option.

Step 1: Begin by using the cosine rule to find the largest angle.

NOTE: We find the largest angle first, because there can only be one angle in a triangle that is obtuse (greater than 90°). If a triangle has an obtuse angle, then this will be it. The reason for finding it first is that in the next step we will use the sine rule to find the second angle. The inverse sin operation that we will use can only give us acute angles (less than 90°), so we avoid a possible wrong answer by first eliminating the only possibility of an obtuse angle.

The largest angle is always opposite to the largest side, so this is angle 'C' in this example ...

The cosine rule...

Find the inverse cos of -0.25 using a scientific calculator...

C = cos-1(-0.25)

= 104.478º

Step 2: Use the sine rule to find one of the remaining angles.

NOTE: The sine rule is easier to use than the cosine rule.

To find angle 'A' with the sine rule:

Find the inverse sin of 0.484123 using a scientific calculator...

A = sin-1(0.484123)

= 28.955º

Step 3: Use the 'sum of internal angles' rule to find the third angle...

The sum of the internal angles equals 180º ...

A + B + C = 180º

so

B = 180º - (A+C)

=  180º - (28.955º + 104.478º)

=  180º - 133.433º

=  46.5675º

The triangle is now solved. This diagram shows all of the sides and angles:

The Method section below shows you how Maths Helper Plus can easily solve your triangles, creating both a labelled diagram and full working steps.

### Method:

Maths Helper Plus can solve a triangle given three side lengths. Full working steps and a labelled diagram are created. The steps below will show you how...

NOTE: This document has already been set up to solve the example triangle as described in the 'theory' section of this topic.

#### Step 2  Display the triangle solver options box

Double click the mouse in the border to the left of the calculations. ( This area is shaded pale blue in the diagram below.) The triangle solver options box will display its 'Lengths & Angles' tab...

Click the 'Clear' button to remove the previous triangle, then click on the 'a' edit box. Now type the length for side 'a' of your triangle. Repeat for 'b' and 'c'.

Click the 'Apply' button at the bottom of the edit box. The calculated values will display on the options box.

Click the 'OK' button to close the options box. The calculations and triangle diagram will be displayed on your screen.

#### Step 3  Adjust the size of the diagram

If the triangle diagram is too big to display properly on your computer screen, briefly press the F10 key to reduce its size. To make the diagram bigger, hold down a Ctrl key while you press F10.