Special Right Triangles Math Example 4

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Example 4

hard
An equilateral triangle has side length 12. Find its height and area using the 30-60-90 triangle relationship.

Solution

  1. 1
    Step 1: An altitude from one vertex of an equilateral triangle bisects the opposite side, creating two 30-60-90 triangles. The hypotenuse of each is 12, and the short leg is 12/2=612/2 = 6.
  2. 2
    Step 2: The height equals the long leg of the 30-60-90 triangle: h=63h = 6\sqrt{3}.
  3. 3
    Step 3: Area of the equilateral triangle =12×base×height=12×12×63=363≈62.35= \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 6\sqrt{3} = 36\sqrt{3} \approx 62.35.

Answer

Height =63= 6\sqrt{3}; Area =363≈62.35= 36\sqrt{3} \approx 62.35 square units.
Dropping an altitude in an equilateral triangle creates two congruent 30-60-90 triangles. The short leg is half the side length, and the long leg (the altitude) is 32\frac{\sqrt{3}}{2} times the side length. This is why the area formula for an equilateral triangle with side ss is s234\frac{s^2\sqrt{3}}{4}.

About Special Right Triangles

Two families of right triangles whose side ratios can be determined exactly: the 30-60-90 triangle with sides in ratio 1:3:21 : \sqrt{3} : 2, and the 45-45-90 triangle with sides in ratio 1:1:21 : 1 : \sqrt{2}.

Learn more about Special Right Triangles →

More Special Right Triangles Examples

Example 1 easy

A 45-45-90 triangle has legs of length 7. Find the length of the hypotenuse.

Example 2 medium

In a 30-60-90 triangle, the hypotenuse is 16. Find the lengths of both legs.

Example 3 easy

The diagonal of a square is 10 cm. Find the side length of the square.

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