Special Right Triangles

By

Special right triangles are 45-45-90 and 30-60-90 triangles, named for their angles. Their sides always follow fixed ratios. In a 45-45-90 triangle, the sides are x, x and x\sqrt{2}. In a 30-60-90 triangle, they are x, x\sqrt{3} and 2x. Use these patterns to find missing sides quickly.

Before this lesson: You should know the Pythagorean Theorem, Isosceles and Equilateral Triangles and Simplifying Square Roots.

Where the Special Right Triangles Come From

Cut a square along its diagonal, and you get two 45-45-90 triangles. If the square’s sides are 1, the Pythagorean theorem gives a diagonal of \sqrt{1^2 + 1^2} = \sqrt{2}.

Cut an equilateral triangle in half with its altitude, and you get two 30-60-90 triangles. If the sides are 2, the base splits into two parts of 1. The altitude is \sqrt{2^2 - 1^2} = \sqrt{3}.

Left: a square with side 1 cut along its diagonal into two 45-45-90 triangles. The diagonal is the square root of 2. Right: an equilateral triangle with side 2 cut by its altitude into two 30-60-90 triangles. The base is split into two segments of 1, and the altitude is the square root of 3.
A 45-45-90 triangle is half a square. A 30-60-90 triangle is half an equilateral triangle.

The Special Right Triangle Ratios

Two right triangles. Left: a 45-45-90 triangle with two legs labeled x and the hypotenuse labeled x times the square root of 2. Both acute angles are 45 degrees. Right: a 30-60-90 triangle. The short leg, across from the 30 degree angle, is x. The long leg, across from the 60 degree angle, is x times the square root of 3. The hypotenuse is 2x.
In a 45-45-90 triangle, the hypotenuse is the leg times √2. In a 30-60-90 triangle, the hypotenuse is twice the short leg, and the long leg is the short leg times √3.

45-45-90 triangle: the two legs are equal, and the hypotenuse is a leg times \sqrt{2}.

\text{leg} : \text{leg} : \text{hypotenuse} = x : x : x\sqrt{2}

30-60-90 triangle: the short leg is across from the 30° angle. The hypotenuse is twice the short leg, and the long leg is the short leg times \sqrt{3}.

\text{short leg} : \text{long leg} : \text{hypotenuse} = x : x\sqrt{3} : 2x

How to Solve Special Right Triangles

  1. Decide which special triangle you have from its angles.
  2. Label the side you know with its part of the ratio: x, x\sqrt{2}, x\sqrt{3} or 2x.
  3. Solve for x.
  4. Use x to find the other sides.
  5. Simplify radicals, and rationalize any denominator.

Worked Examples

Example 1: 45-45-90 triangle, given a leg

A 45-45-90 triangle has a leg of 7. Find the other leg and the hypotenuse.

  1. The legs are equal: the other leg is 7.
  2. The hypotenuse is a leg times \sqrt{2}: 7\sqrt{2}
  3. Use a calculator and round: 7\sqrt{2} \approx 9.90

Answer: The other leg is 7, and the hypotenuse is 7\sqrt{2} \approx 9.90.

Example 2: 45-45-90 triangle, given the hypotenuse

A 45-45-90 triangle has a hypotenuse of 10. Find each leg.

  1. Set the hypotenuse equal to x\sqrt{2}: x\sqrt{2} = 10
  2. Divide both sides by \sqrt{2}: x = \tfrac{10}{\sqrt{2}}
  3. Multiply the top and bottom by \sqrt{2} (see rationalizing the denominator): x = \tfrac{10\sqrt{2}}{2}
  4. Simplify: x = 5\sqrt{2}
  5. Use a calculator and round: 5\sqrt{2} \approx 7.07

Answer: Each leg is 5\sqrt{2} \approx 7.07.

Example 3: 30-60-90 triangle, given the short leg

A 30-60-90 triangle has a short leg of 5. Find the long leg and the hypotenuse.

  1. The short leg is x: x = 5
  2. The hypotenuse is 2x: 2 \cdot 5 = 10
  3. The long leg is x\sqrt{3}: 5\sqrt{3}
  4. Use a calculator and round: 5\sqrt{3} \approx 8.66

Answer: The hypotenuse is 10, and the long leg is 5\sqrt{3} \approx 8.66.

Example 4: 30-60-90 triangle, given the long leg

A 30-60-90 triangle has a long leg of 9. Find the short leg and the hypotenuse.

  1. Set the long leg equal to x\sqrt{3}: x\sqrt{3} = 9
  2. Divide both sides by \sqrt{3}: x = \tfrac{9}{\sqrt{3}}
  3. Multiply the top and bottom by \sqrt{3}: x = \tfrac{9\sqrt{3}}{3}
  4. Simplify: x = 3\sqrt{3}
  5. The hypotenuse is 2x: 2 \cdot 3\sqrt{3} = 6\sqrt{3}

Answer: The short leg is 3\sqrt{3} \approx 5.20, and the hypotenuse is 6\sqrt{3} \approx 10.39.

Example 5: Word problem with a yield sign

A yield sign is an equilateral triangle with 30-inch sides. How tall is the sign?

  1. The altitude splits the sign into two 30-60-90 triangles.
  2. The hypotenuse of each half is a side of the sign: 2x = 30
  3. Divide both sides by 2: x = 15
  4. The height is the long leg, x\sqrt{3}: 15\sqrt{3}
  5. Use a calculator and round: 15\sqrt{3} \approx 25.98

Answer: The sign is 15\sqrt{3} \approx 26 inches tall. Its area, from area of triangles, is \tfrac{1}{2} \cdot 30 \cdot 15\sqrt{3} = 225\sqrt{3} \approx 389.7 square inches.

Common Mistakes

Mistake Example of the error Fix
Mixing up the long leg and the hypotenuse Short leg 5, so “hypotenuse = 5\sqrt{3}” The hypotenuse is the longest side, 2x = 10. The long leg is 5\sqrt{3} \approx 8.66.
Putting x across from the 60° angle Calling the side across from 60° the short leg The short leg is always across from the 30° angle.
Doubling the long leg Long leg 5\sqrt{3}, so hypotenuse = 10\sqrt{3} Double the short leg, not the long leg: 2 \cdot 5 = 10.
Using 2x for a 45-45-90 hypotenuse Leg 7, so hypotenuse = 14 In a 45-45-90 triangle, the hypotenuse is x\sqrt{2}: 7\sqrt{2}.
Leaving a radical in the denominator Leg = \tfrac{10}{\sqrt{2}} Rationalize: \tfrac{10}{\sqrt{2}} = 5\sqrt{2}.

Special Right Triangles at a Glance

Triangle Given To find the others
45-45-90 A leg x Other leg x, hypotenuse x\sqrt{2}
45-45-90 Hypotenuse h Each leg \tfrac{h}{\sqrt{2}} = \tfrac{h\sqrt{2}}{2}
30-60-90 Short leg x Long leg x\sqrt{3}, hypotenuse 2x
30-60-90 Hypotenuse h Short leg \tfrac{h}{2}, long leg \tfrac{h}{2}\sqrt{3}
30-60-90 Long leg L Short leg \tfrac{L}{\sqrt{3}} = \tfrac{L\sqrt{3}}{3}, hypotenuse twice that

You can always check an answer with the Pythagorean theorem. These ratios also give the exact values of the trig ratios for 30°, 45° and 60°.

Practice Problems

  1. A 45-45-90 triangle has a leg of 4. Find the hypotenuse.
  2. A 45-45-90 triangle has a hypotenuse of 6\sqrt{2}. Find each leg.
  3. A 45-45-90 triangle has a hypotenuse of 8. Find each leg in simplest radical form.
  4. A 30-60-90 triangle has a short leg of 3. Find the long leg and the hypotenuse.
  5. A 30-60-90 triangle has a hypotenuse of 20. Find the short leg and the long leg.
  6. A 30-60-90 triangle has a long leg of 6\sqrt{3}. Find the short leg and the hypotenuse.
  7. A 30-60-90 triangle has a long leg of 12. Find the short leg and the hypotenuse in simplest radical form.
  8. A square has sides of 9 cm. How long is its diagonal? Give the exact answer and round to the nearest hundredth.
  9. An equilateral triangle has sides of 10. Find its height and its area in simplest radical form.
  10. A 16-foot ladder leans against a wall and makes a 60° angle with the ground. How high up the wall does it reach, and how far is its base from the wall?
Show answers
  1. Hypotenuse is a leg times \sqrt{2}: 4\sqrt{2}
  2. Set up: x\sqrt{2} = 6\sqrt{2}
    Divide by \sqrt{2}: x = 6
    Each leg is 6.
  3. Set up: x\sqrt{2} = 8
    Divide by \sqrt{2}: x = \tfrac{8}{\sqrt{2}}
    Rationalize: x = \tfrac{8\sqrt{2}}{2}
    Simplify: x = 4\sqrt{2}
  4. Short leg: x = 3
    Hypotenuse: 2 \cdot 3 = 6
    Long leg: 3\sqrt{3}
  5. Set up: 2x = 20
    Divide by 2: x = 10
    Short leg is 10.
    Long leg: 10\sqrt{3}
  6. Set up: x\sqrt{3} = 6\sqrt{3}
    Divide by \sqrt{3}: x = 6
    Short leg is 6.
    Hypotenuse: 2 \cdot 6 = 12
  7. Set up: x\sqrt{3} = 12
    Divide by \sqrt{3}: x = \tfrac{12}{\sqrt{3}}
    Rationalize: x = \tfrac{12\sqrt{3}}{3}
    Simplify: x = 4\sqrt{3}
    Hypotenuse: 2 \cdot 4\sqrt{3} = 8\sqrt{3}
  8. The diagonal cuts the square into two 45-45-90 triangles with legs of 9.
    Diagonal: 9\sqrt{2}
    Round: 9\sqrt{2} \approx 12.73 cm
  9. Each half is a 30-60-90 triangle with hypotenuse 10.
    Set up: 2x = 10
    Divide by 2: x = 5
    Height is the long leg: 5\sqrt{3}
    Area: \tfrac{1}{2} \cdot 10 \cdot 5\sqrt{3}
    Multiply: 25\sqrt{3} \approx 43.30
  10. The ladder is the hypotenuse: 2x = 16
    Divide by 2: x = 8
    The base is across from the 30° angle at the top, so it is the short leg: 8 ft.
    The height is across from the 60° angle, so it is the long leg: 8\sqrt{3} \approx 13.86 ft.

FAQ

What are special right triangles?

They are right triangles with angles of 45°, 45°, 90° or 30°, 60°, 90°. Their sides always follow fixed ratios, so you can find missing sides without the Pythagorean theorem.

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

The sides are in the ratio x : x : x\sqrt{2}. The two legs are equal, and the hypotenuse is a leg times \sqrt{2}.

What is the ratio of a 30-60-90 triangle?

The sides are in the ratio x : x\sqrt{3} : 2x. These are the short leg, the long leg and the hypotenuse.

How do you find the legs of a 45-45-90 triangle from the hypotenuse?

Divide the hypotenuse by \sqrt{2}, then rationalize. For a hypotenuse of 10, each leg is \tfrac{10}{\sqrt{2}} = 5\sqrt{2}.

Summary

  • 45-45-90: x : x : x\sqrt{2}. It is half a square.
  • 30-60-90: x : x\sqrt{3} : 2x. It is half an equilateral triangle.
  • The short leg of a 30-60-90 triangle is across from the 30° angle.
  • Solve for x first, then find the other sides.
  • Rationalize when you divide by \sqrt{2} or \sqrt{3}.

Previous lesson: Converse of the Pythagorean Theorem

Next lesson: Similar Right Triangles

Back to the Geometry Index