Math Core

If someone gives you three side lengths or a few angles, can you always build a triangle from them? Sometimes you can build exactly one, sometimes many, and sometimes none at all. Knowing which is which is the key to drawing triangles from given conditions.

Building a triangle from three sides

Imagine three straws cut to $3$ cm, $4$ cm and $5$ cm. Lay the $4$ cm straw down, then swing the other two up from its ends until they touch. They meet at one point, and you get a triangle.

Straws of 3 cm, 4 cm and 5 cm meet to form a triangle.

Now try straws of $2$ cm, $3$ cm and $7$ cm. Lay the $7$ cm straw down. The $2$ cm straw can only reach points on a circle of radius $2$ around the left end, and the $3$ cm straw can only reach a circle of radius $3$ around the right end. Those circles never touch, so the straws can't meet.

The 2 cm and 3 cm straws can reach only the points on these circles. The circles don't meet, so there is no triangle.

The two short straws together are only $2 + 3 = 5$ cm long, which is shorter than the $7$ cm side. They fall short no matter how you tilt them.

The triangle inequality

Three lengths make a triangle only when the two shorter lengths add up to more than the longest length.

$3, 4, 5$ : $3 + 4 = 7 > 5$ , so a triangle works.
$2, 3, 7$ : $2 + 3 = 5 < 7$ , so no triangle.
$3, 4, 7$ : $3 + 4 = 7$ , exactly equal. The straws lie flat along the long one, so no triangle.

When three lengths do work, they make exactly one triangle. You might flip it or turn it, but the shape and size are always the same.

Worked example: Which sets make a triangle?

Do these lengths make a triangle? (a) $6, 8, 13$ (b) $5, 9, 15$

(a) The two shorter sides: $6 + 8 = 14$ , and $14 > 13$ . Yes.

(b) The two shorter sides: $5 + 9 = 14$ , and $14 < 15$ . No.

Worked example: Possible third sides

Two sides of a triangle are $5$ and $9$ . What lengths could the third side be?

If the third side is the longest, it must be less than $5 + 9 = 14$ . If $9$ is the longest, then the third side plus $5$ must be more than $9$ , so the third side is more than $9 - 5 = 4$ . The third side must be between $4$ and $14$ .

Building a triangle from angles

The three angles of every triangle add up to $180^\circ$ . So angles of $50^\circ$ , $60^\circ$ and $70^\circ$ can make a triangle ( $50 + 60 + 70 = 180$ ), but $50^\circ$ , $60^\circ$ and $80^\circ$ cannot.

Even when the angles work, they don't fix the size. A tiny triangle and a huge triangle can both have angles of $50^\circ$ , $60^\circ$ and $70^\circ$ . They have the same shape but different sizes, so three angles give more than one triangle.

How many triangles?

Here is a summary of what different conditions give you.

Given	Number of triangles
Three sides that pass the triangle inequality	exactly one
Two sides and the angle between them	exactly one
Two angles and the side between them	exactly one
Three angles that add to $180^\circ$	more than one (same shape, any size)
Sides that fail the triangle inequality, or angles that don't add to $180^\circ$	none

Worked example: Two sides and the angle between

Can you draw a triangle with sides $4$ cm and $6$ cm and a $40^\circ$ angle between them? How many?

Draw the $6$ cm side. At one end, use a protractor to draw a $40^\circ$ angle, and mark $4$ cm along that ray. Connecting the two free endpoints closes the triangle. Every choice was forced, so there is exactly one triangle.

Common mistake

Always add the two shorter sides and compare with the longest. For $4, 10, 5$ , checking $4 + 10 > 5$ tells you nothing. The test that matters is $4 + 5 = 9$ , which is less than $10$ , so there's no triangle.

Practice

Practice 1

Can a triangle have sides of $4$ cm, $5$ cm and $10$ cm?

Practice 2

Which set of lengths makes a triangle?

Practice 3

Two angles of a triangle measure $48^\circ$ and $67^\circ$ . What is the measure of the third angle, in degrees?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

How many different triangles have angles of $35^\circ$ , $65^\circ$ and $80^\circ$ ?

Practice 5

How many different triangles have sides of $5$ cm and $8$ cm with a $70^\circ$ angle between those two sides?

Practice 6

A triangle has two sides of length $5$ . The third side is a whole number. What is the largest it can be?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Two sides of a triangle are $6$ and $10$ . The third side is a whole number. How many different lengths are possible for the third side?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.