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45 45 90 Triangle Calculator

Calculate all sides and angles of a 45-45-90 special right triangle. Enter any side length to find the remaining measurements instantly.

What this tool does

Calculate all sides and angles of a 45-45-90 special right triangle. Enter any side length to find the remaining measurements instantly. You fill in these fields: Known side and Length of the known side (any unit). It shows hypotenuse, leg 1, leg 2, area, perimeter, and each acute angle.

Inputs

  • Known side: Choose one: Leg (either one), Hypotenuse. Starts as Leg (either one).
  • Length of the known side (any unit) (units): A number. Must be greater than 0.

How to use

  1. Choose Leg (either one) or Hypotenuse for "Known side".
  2. Enter the length of the known side (any unit) in units.
  3. Read the result: hypotenuse, leg 1, leg 2, area, perimeter, and each acute angle. It appears once every required field has a valid value.

Worked examples

Example 1. Inputs:

  • Known side: Leg (either one)
  • Length of the known side (any unit): 10 units

Results:

  • Hypotenuse: 14.142136 units
  • Leg 1: 10 units
  • Leg 2: 10 units
  • Area: 50 square units
  • Perimeter: 34.142136 units
  • Each acute angle: 45 degrees

How this is worked out: Known side is a leg, so both legs are 10 and the hypotenuse is 10 x sqrt(2) = 14.1421356, shown as 14.142136. Area = 10 x 10 / 2 = 50. Perimeter = 10 + 10 + 14.1421356 = 34.1421356, shown as 34.142136. Equal legs give equal acute angles that sum to 90, so each is 45 degrees.

Example 2. Inputs:

  • Known side: Hypotenuse
  • Length of the known side (any unit): 10 units

Results:

  • Hypotenuse: 10 units
  • Leg 1: 7.071068 units
  • Leg 2: 7.071068 units
  • Area: 25 square units
  • Perimeter: 24.142136 units
  • Each acute angle: 45 degrees

How this is worked out: Known side is the hypotenuse, so each leg is 10 / sqrt(2) = 10 x 0.70710678 = 7.0710678, shown as 7.071068. Area = 7.0710678 x 7.0710678 / 2 = 50 / 2 = 25. Perimeter = 7.0710678 + 7.0710678 + 10 = 24.1421356, shown as 24.142136. Angles are 45 degrees by the same symmetry.

Example 3. Inputs:

  • Known side: Leg (either one)
  • Length of the known side (any unit): 3 units

Results:

  • Hypotenuse: 4.242641 units
  • Leg 1: 3 units
  • Leg 2: 3 units
  • Area: 4.5 square units
  • Perimeter: 10.242641 units
  • Each acute angle: 45 degrees

How this is worked out: Known leg of 3: hypotenuse = 3 x sqrt(2) = 3 x 1.41421356 = 4.24264069, shown as 4.242641. Area = 3 x 3 / 2 = 4.5. Perimeter = 3 + 3 + 4.24264069 = 10.24264069, shown as 10.242641. Angles 45 degrees.

Example 4. Inputs:

  • Known side: Hypotenuse
  • Length of the known side (any unit): 1 unit

Results:

  • Hypotenuse: 1 units
  • Leg 1: 0.707107 units
  • Leg 2: 0.707107 units
  • Area: 0.25 square units
  • Perimeter: 2.414214 units
  • Each acute angle: 45 degrees

How this is worked out: Known hypotenuse of 1: each leg = 1 / sqrt(2) = 0.70710678, shown as 0.707107. Area = 0.70710678 x 0.70710678 / 2 = 0.5 / 2 = 0.25. Perimeter = 0.70710678 x 2 + 1 = 2.41421356, shown as 2.414214. Angles 45 degrees.

Limitations

  • Only for 45-45-90 triangles, which have two equal legs. Other right triangles give wrong results.
  • Enter a length greater than 0. Zero and negative lengths are refused.
  • Results are rounded for display, so the hypotenuse may differ very slightly from an exact value.

FAQs

Q: Which side should I enter?
A: Enter the length of one leg, or the length of the hypotenuse, and choose which one you entered. The two legs of this triangle are always equal.
Q: How is the hypotenuse found from a leg?
A: The hypotenuse is the leg multiplied by the square root of 2, which is about 1.41421 times the leg.
Q: How is a leg found from the hypotenuse?
A: Each leg is the hypotenuse divided by the square root of 2, which is about 0.70711 times the hypotenuse.
Q: Why are the acute angles always 45 degrees?
A: The two legs are equal, so the two acute angles are equal. Together they make 90 degrees, so each one is 45 degrees.

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