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Geometry

Special Right Triangles

Lesson

Two right triangles are special because their side ratios come out exact (without messy decimals). Memorize them and you never have to use the Pythagorean theorem again for these.

45–45–90 (isosceles right)

legs:leg:hyp=1:1:2\text{legs} : \text{leg} : \text{hyp} = 1 : 1 : \sqrt{2}

Both legs are equal. Multiply a leg by 2\sqrt{2} to get the hypotenuse.

30–60–90

short leg:long leg:hyp=1:3:2\text{short leg} : \text{long leg} : \text{hyp} = 1 : \sqrt{3} : 2
  • Short leg is opposite the 30∘30^\circ.
  • Hypotenuse = 2 × short leg.
  • Long leg = short leg × 3\sqrt{3}.

Worked example — 30-60-90

A 30-60-90 has short leg 7. Hypotenuse:

2⋅7=142 \cdot 7 = 14

For coefficient questions

If a problem asks for a long leg or a hypotenuse that involves 2\sqrt{2} or 3\sqrt{3}, type the coefficient (the number in front of the root). So if the long leg of a 30-60-90 is 636\sqrt{3}, type 6.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

30-60-90: short leg 5. Hypotenuse?\text{30-60-90: short leg 5. Hypotenuse?}

Problem 2

30-60-90: hypotenuse 14. Short leg?\text{30-60-90: hypotenuse 14. Short leg?}

Problem 3

45-45-90: one leg 7. Other leg?\text{45-45-90: one leg 7. Other leg?}

Problem 4

30-60-90: short leg 4. Long-leg coefficient (⋅3)?\text{30-60-90: short leg 4. Long-leg coefficient (}{} \cdot \sqrt{3})?

Practice

Standard problems matching the lesson.

Problem 5

30-60-90: short leg 8. Hyp?\text{30-60-90: short leg 8. Hyp?}

Problem 6

30-60-90: hyp 30. Short leg?\text{30-60-90: hyp 30. Short leg?}

Problem 7

45-45-90: leg 15. Other leg?\text{45-45-90: leg 15. Other leg?}

Problem 8

30-60-90: short 6. Long coefficient?\text{30-60-90: short 6. Long coefficient?}

Problem 9

30-60-90: short 10. Long coefficient?\text{30-60-90: short 10. Long coefficient?}

Problem 10

30-60-90: hyp 24. Short?\text{30-60-90: hyp 24. Short?}

Problem 11

30-60-90: hyp 100. Short?\text{30-60-90: hyp 100. Short?}

Problem 12

45-45-90: leg 20. Other?\text{45-45-90: leg 20. Other?}

Problem 13

30-60-90: short 1. Hyp?\text{30-60-90: short 1. Hyp?}

Problem 14

30-60-90: hyp 2. Short?\text{30-60-90: hyp 2. Short?}

Problem 15

30-60-90: short 11. Hyp?\text{30-60-90: short 11. Hyp?}

Problem 16

30-60-90: hyp 50. Short?\text{30-60-90: hyp 50. Short?}

Problem 17

Square diagonal 82. Side?\text{Square diagonal } 8\sqrt{2}. \text{ Side?}

Problem 18

Square side 6. Diagonal coefficient\text{Square side 6. Diagonal coefficient}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

30-60-90: hyp 18. Short?\text{30-60-90: hyp 18. Short?}

Problem 20

30-60-90: short 12. Hyp?\text{30-60-90: short 12. Hyp?}

Problem 21

45-45-90: hyp 10. Leg coefficient (⋅2)?\text{45-45-90: hyp 10. Leg coefficient (}{} \cdot \sqrt{2})?

Problem 22

30-60-90: short 2. Long coefficient?\text{30-60-90: short 2. Long coefficient?}

Problem 23

30-60-90: hyp 6. Long-leg coefficient?\text{30-60-90: hyp 6. Long-leg coefficient?}

Problem 24

Equilateral side 8. Height coefficient (⋅3)?\text{Equilateral side 8. Height coefficient (}{} \cdot \sqrt{3})?

Problem 25

30-60-90: short 15. Hyp?\text{30-60-90: short 15. Hyp?}

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