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What Are 30°–60°–90° Triangles?

When you have a triangle with the angles 30°, 60°, and 90°, the length of the shorter leg is always half the length of the hypotenuse—the longest side, longer than the two legs. You also know that the longer leg is always equal to 3 multiplied by the length of the shorter leg.

30, 60, 90 triangle with the sides 2k, k and square root of 3k

Rule

In a “30°-60°-90°” triangle, the sides have the following relationships: short leg = k long leg = 3k hypotenuse = 2k

where 3 1.73.

Example 1

A 30°-60°-90° triangle has a hypotenuse of length 12cm. Find the length of the legs.

Since this is a “30°-60°-90°” triangle, you already know that the shorter leg measures half the length of the hypotenuse, or k = 12cm 2 = 6cm. You can now find the lengtt of the longer leg either by using the Pythagorean theorem, or by taking 3k. Here I’ll show you both of them.

First, the Pythagorean theorem: k2 + 62 = 122 k2 = 144 36 k2 = 108 k 10.39

With 3k: long leg = 3 k 3 6cm 10.39cm

The longer leg is about 10.39 cm long.

Example 2

You have a 30°-60°-90° triangle where the longer leg measures 3cm. Find the length of the two other sides.

Since this is a “30°-60°-90°” triangle, you know that the shorter leg’s length is half of the hypotenuse h. Therefore you can make the following equation using the Pythagorean theorem: 32 + (1 2h)2 = h2 9 = h2 1 4h2 9 = 3 4 h2| 4 3 4 3 9 = h2 12 = h2 3.5 h

The hypotenuse measures about 3.5 cm. The shorter leg is half of the hypotenuse, so it is

k 3.5cm 2 1.75cm

Example 3

You have an equilateral triangle where all sides are 6cm. What is the height of the triangle?

Because the angles in an equilateral triangle are all 60°, you can find the height of the triangle by using the formula for a “30°-60°-90°” triangle. The height becomes one of the legs in a “30°-60°-90°” triangle:

Example of 30, 60, 90 triangle with side length 6 cm

As you can see, the hypotenuse is 6 cm, and the shorter leg of the new triangle is exactly half of the hypotenuse (and half of the length of the side of the original triangle), which is 3 cm.

You then calculate: h2 + 32 = 62 h2 = 36 9 h2 = 27 h 5.2

The height h is therefore about 5.2 cm.

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