The 30-60-90 Triangle
Grade 8 math lesson on the 30-60-90 special right triangle and its side ratios. Students learn the fixed ratio 1 : square root of 3 : 2 between the sides of a 30-60-90 triangle and use it to find missing side lengths without using the Pythagorean Theorem.
Key Concepts
Property A 30 60 90 triangle is half of an equilateral triangle. All 30 60 90 triangles have side lengths in the ratio of $1:\sqrt{3}:2$.
Examples If the shortest side of a 30 60 90 triangle is 5 inches, the hypotenuse is $2 \times 5 = 10$ inches. In that same triangle, the side opposite the 60° angle would be $5\sqrt{3}$ inches long. An equilateral triangle with 12 inch sides has a height of $6\sqrt{3}$ inches, found by splitting it into two 30 60 90 triangles.
Explanation Picture an equilateral triangle with all its equal sides. Now, chop it right down the middle! You've just created two 30 60 90 triangles. The shortest leg is always opposite the 30 degree angle, the hypotenuse is simply double the short leg, and the medium leg is the short leg's length times the square root of three.
Common Questions
What is a 30-60-90 triangle?
A 30-60-90 triangle is a special right triangle with angles of 30 degrees, 60 degrees, and 90 degrees. Its sides are always in the ratio 1 : square root of 3 : 2, which allows you to find any missing side if you know one side.
What are the side ratios of a 30-60-90 triangle?
If the shortest side (opposite the 30-degree angle) has length x, then the side opposite the 60-degree angle has length x times the square root of 3, and the hypotenuse (opposite the 90-degree angle) has length 2x.
How do you find a missing side in a 30-60-90 triangle?
Identify which angle is opposite the known side. Use the 1 : root 3 : 2 ratio to scale up or down. If the hypotenuse is 10, then the shortest side is 5 (half) and the middle side is 5 times root 3.
Where do 30-60-90 triangles appear in geometry?
These triangles appear when you draw the altitude of an equilateral triangle, in hexagonal patterns, and in many construction and engineering problems. Knowing the ratios speeds up many geometric calculations.