Cal11 calculator
Search calculator...

Pythagorean Calculator 45 Degrees

Reviewed by Calculator Editorial Team

The Pythagorean Calculator 45 Degrees helps you determine the sides of a right triangle when one of the non-right angles is 45 degrees. This special right triangle has unique properties that make calculations simpler than with arbitrary right triangles.

What is the Pythagorean Theorem?

The Pythagorean Theorem is a fundamental principle in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Pythagorean Theorem Formula

a² + b² = c²

Where:

  • a and b are the lengths of the legs
  • c is the length of the hypotenuse

This theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery. It's one of the most important theorems in mathematics with applications in many fields including engineering, architecture, and physics.

Special Right Triangle: 45-Degree Angle

A right triangle with one 45-degree angle is a special case that has specific properties:

  • It's an isosceles right triangle (the two non-right angles are equal)
  • The two legs are equal in length
  • The hypotenuse is √2 times the length of each leg

Key Property

In a 45-45-90 triangle, the hypotenuse is always √2 times longer than either leg.

This property makes calculations with 45-degree triangles particularly straightforward compared to other right triangles.

How to Use This Calculator

  1. Enter the length of one side of the triangle in the input field
  2. Select whether you're providing a leg or the hypotenuse
  3. Click "Calculate" to see the results
  4. View the complete solution and visual representation

The calculator will show you all sides of the triangle based on your input, along with a visual representation of the triangle.

Formula

For a 45-45-90 Triangle

If you know one leg (a):

Other leg = a

Hypotenuse = a√2

If you know the hypotenuse (c):

Leg = c/√2

These formulas are derived from the general Pythagorean theorem and the specific properties of 45-45-90 triangles.

Worked Example

Example Calculation

Suppose you have a right triangle with one 45-degree angle and one leg measuring 5 units.

Using the formula:

  • Other leg = 5 units
  • Hypotenuse = 5 × √2 ≈ 7.071 units

This means the other leg is also 5 units, and the hypotenuse is approximately 7.071 units.

This example demonstrates how the properties of a 45-45-90 triangle simplify the calculations compared to arbitrary right triangles.

Frequently Asked Questions

What is a 45-45-90 triangle?
A 45-45-90 triangle is a right triangle where the two non-right angles are both 45 degrees. It's also an isosceles triangle with two equal sides.
How do I calculate the sides of a 45-45-90 triangle?
If you know one leg, the other leg is equal, and the hypotenuse is √2 times the leg length. If you know the hypotenuse, each leg is the hypotenuse divided by √2.
What's the difference between a 45-45-90 triangle and a 30-60-90 triangle?
A 45-45-90 triangle has two equal legs and a hypotenuse that's √2 times a leg. A 30-60-90 triangle has sides in the ratio 1 : √3 : 2.