How to solve a triangle given two sides and the angle between them.





Part 1

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 unknown side and angles when any two sides and the included angle are given.

NOTE: The included angle is the angle between the two given sides.

These are the formulas used to solve triangles:

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

A + B + C = 180º

    1. The ‘sine rule‘ …

    1. The ‘cosine rule‘ …

           a² = b² + c² – 2bc cosA


             b² = a² + c² – 2ac cosB


             c² = b² + a² – 2ba cosC

We will now use an example to show how these rules are applied to solve a triangle when two sides and the included angle are given.

Example: Solve this triangle for the unknown side, ‘a’, and internal angles ‘B’ and ‘C’:

Step 1: Begin by using the cosine rule to find the unknown side.

The cosine rule…

a² = b² + c² – 2bc cosA

            = 3² + 4² – 2×3×4 cos30°

            = 4.215390309

Taking the positive square root…

         a = 2.05314

Step 2: Use the sine rule to find the smaller of the two unknown angles.

NOTE: We find the smaller angle first because we are about to use the sine rule. 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 making sure the angle to be found is acute. There can only be one angle in a triangle that is obtuse (greater than 90°), and it would always be opposite the largest side. To choose the smaller of the remaining angles, look at the sides opposite to them. A smaller opposite side means a smaller angle.

For the example triangle, side ‘b’ is smaller than side ‘c’, so angle B must be smaller than angle C. We choose to find angle B with the sine rule.

To find angle ‘B’ with the sine rule:

Find the inverse sin of 0.730588 using a scientific calculator…

B = sin-1(0.730588)

            = 46.9357º

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


        C = 180º – (A+B) 

           =  180 – (30º + 46.9357º)

           =  180 – 76.9357º

           =   103.064º

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 labeled diagram and full working steps.

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Part 2

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…

Step 1 Download the free support file…We have created a Maths Helper Plus document containing the completed example from this topic. You can use this to practice the steps described below, and as a starting point for solving your own problems.

File name: ‘Triangle sover – SAS.mhp’ File size: 4kb

If you choose` ‘Open this file from its current location’, then Maths Helper Plus should open the document immediately. If not, try the other option: ‘Save this file to disk’, then run Maths Helper Plus and choose the ‘Open’ command from the ‘File’ menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.

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 enter your given angle and two sides. You can enter these in three ways, but the answer will be correct as long as the given angle is between the given sides.

In the diagram above, we are entering sides ‘b’ and ‘c’, and the angle ‘A’ which is between ‘b’ and ‘c’.

To enter a value, click on its edit box, then type the value.

NOTE: Angles are already in degrees. Do not follow with a degree symbol: °

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.

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