Practice Special Right Triangles in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

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}.

Cut an equilateral triangle in half and you get a 30-60-90 triangle. Cut a square along its diagonal and you get a 45-45-90 triangle. These two cuts give you exact side ratios you can memorize forever—no calculator needed.

Showing a random 20 of 50 problems.

Example 1

challenge
Verify the 30-60-90 ratio 1:3:21 : \sqrt 3 : 2 satisfies the Pythagorean theorem.

Example 2

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

Example 3

challenge
A 45-45-90 triangle is inscribed so its hypotenuse is the diameter of a circle of radius 5. Find the legs.

Example 4

challenge
A regular octagon is inscribed in a circle of radius 11. Find the length of one side exactly.

Example 5

easy
sin⁡30°=\sin 30° = ___.

Example 6

easy
A 30-60-90 triangle has its shortest side (opposite 30°) equal to 4. Find the hypotenuse.

Example 7

easy
A 30-60-90 triangle has its short leg =6= 6. Find the long leg.

Example 8

medium
An equilateral triangle has side 6. Find its height exactly.

Example 9

easy
A 45-45-90 triangle has hypotenuse 626\sqrt 2. Find the leg length.

Example 10

easy
A 45-45-90 triangle has legs of length 33. Find the hypotenuse.

Example 11

medium
Find cos⁡60°\cos 60° exactly.

Example 12

hard
A regular hexagon has area 54354\sqrt 3. Find its side length.

Example 13

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

Example 14

challenge
A regular hexagon has side 6. Using special triangles, find its area.

Example 15

hard
A square is inscribed in a circle of radius 55. Find its area.

Example 16

hard
Find the area of an equilateral triangle with perimeter 1818.

Example 17

medium
A ramp rises at 30°30° to a height of 44 ft. Find the ramp length.

Example 18

medium
An equilateral triangle has side 88. Find its altitude exactly.

Example 19

easy
What is the side ratio of a 45-45-90 triangle?

Example 20

hard
An isosceles right triangle has area 5050. Find its hypotenuse.
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