30 60 90 Triangle Worksheet Answer Key

30 60 90 Triangle Worksheet Answer Key – Complete Guide with Practice Problems and Solutions

The 30‑60‑90 triangle is one of the most useful special right triangles in geometry. In practice, its side lengths follow a fixed ratio of 1 : √3 : 2, where the shortest side (opposite the 30° angle) is the shorter leg, the longer leg (opposite the 60° angle) is √3 times the short leg, and the hypotenuse is twice the short leg. In practice, mastering this relationship enables students to solve a wide range of problems quickly and accurately. This article provides a ready‑to‑use 30 60 90 triangle worksheet answer key that includes ten practice questions, detailed step‑by‑step solutions, and explanations of the underlying concepts. Whether you are a teacher preparing materials or a learner seeking extra practice, this resource will help you build confidence in handling special right triangles.

Worksheet Questions

Below are ten problems designed to test understanding of the 30‑60‑90 triangle properties. Each question can be copied into a worksheet or used as individual practice items.

1. In a 30‑60‑90 triangle, the side opposite the 30° angle measures 4 cm. Find the lengths of the other two sides.
2. The hypotenuse of a 30‑60‑90 triangle is 12 units. Determine the lengths of the legs.
3. A 30‑60‑90 triangle has a longer leg of 9 √3 inches. What is the length of the shorter leg?
4. If the shorter leg of a 30‑60‑90 triangle is 5 ft, calculate the perimeter of the triangle.
5. The area of a 30‑60‑90 triangle is 24 cm² and the shorter leg is 6 cm. Find the length of the hypotenuse.
6. In a right triangle, one acute angle measures 30°. The side opposite this angle is 7 units. Identify the other two sides.
7. A 30‑60‑90 triangle is inscribed in a circle with a radius of 5 units. What are the side lengths of the triangle?
8. The longer leg of a 30‑60‑90 triangle is 15 units. Compute the triangle’s area.
9. A 30‑60‑90 triangle has a perimeter of 30 units. Determine the length of each side.
10. The hypotenuse of a 30‑60‑90 triangle is 8 √2 units. Find the lengths of the legs.

Answer Key with Detailed Solutions

1. Shorter leg = 4 cm

• Longer leg = 4 √3 cm
• Hypotenuse = 8 cm

Explanation: Using the ratio 1 : √3 : 2, multiply each part by 4 And that's really what it comes down to..

2. Hypotenuse = 12 units → short leg = 6, long leg = 6√3

• Short leg = 12 ÷ 2 = 6
• Long leg = 6 × √3 = 6√3

3. Longer leg = 9√3 → short leg = 9

• Short leg = (9√3) ÷ √3 = 9

4. Short leg = 5 ft → long leg = 5√3 ft, hypotenuse = 10 ft

• Perimeter = 5 + 5√3 + 10 ≈ 5 + 8.66 + 10 = 23.66 ft (exact form: 15 + 5√3 ft)

5. Area = 24 cm², short leg = 6 cm → height = short leg (since area = ½ × short × long)

• Long leg = (2 × Area) ÷ short = (48) ÷ 6 = 8 cm
• Hypotenuse = 2 × short = 12 cm

6. Opposite 30° = 7 units → short leg = 7, long leg = 7√3, hypotenuse = 14

• Short leg = 7
• Long leg = 7√3
• Hypotenuse = 2 × 7 = 14

7. Radius = 5 → hypotenuse = diameter = 10

• Short leg = 5 (half of hypotenuse)
• Long leg = 5√3
• Hypotenuse = 10

8. Longer leg = 15 → short leg = 15 ÷ √3 = 5√3

• Area = ½ × short × long = ½ × (5√3) × 15 = (75√3)/2 ≈ 64.95 units²

9. Perimeter = 30 → let short leg = x, then sides are x, x√3, 2x

• Equation: x + x√3 + 2x = 30 → x(3 + √3) = 30 → x = 30 ÷ (3 + √3)
• Simplify: Multiply numerator and denominator by (3 − √3):
  x = 30(3 − √3) ÷ (9 − 3) = 30(3 − √3) ÷ 6 = 5(3 − √3) = 15 − 5√3
• Side lengths:
  • Short leg = 15 − 5√3
  • Long leg = (15 − 5√3)√3 = 15√3 − 15
  • Hypotenuse = 2(15 − 5√3) = 30 − 10√3

10. Hypotenuse = 8√2 → short leg = 4√2, long leg