How to solve a triangle given three sides.
Goal:
Theory:
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 three unknown angles when the three side lengths ‘a’, ‘b’ and ‘c’ are known.
These are the formulas used to solve triangles:

 The sum of the internal angles equals 180º …
A + B + C = 180º

 The ‘sine rule‘ …


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:
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 – SSS.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 parameters 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.
Download Free Support File
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