45 45 90 Triangle Calculator

45 45 90 Triangle Calculator

Select the parameter and input its value to calculate all the missing elements of a 45-45-90 triangle.

Side a:

Side (a)

cm ▾

The 45 45 90 triangle calculator solves for all sides, area, perimeter, inradius, and circumradius of special right triangles 45 45 90. Enter any of the parameters of your choice and get calculations for all others in seconds and accurately.

45 45 90 Triangle:

A 45-45-90 triangle is a special type of right triangle in which two interior angles are 45° each, and the third angle is a right angle (90°).

45 45 90 triangle

45-45-90 Triangle Ratios:

Angle Ratios	Side Ratios
45° : 45° : 90° (1 : 1 : 2)	a : a : a√2 (1 : 1 : √2)

45-45-90 Triangle Formulas:

The formulas for solving this triangle are derived from trigonometry. Our 45-45-90 triangle calculator uses the following equations to compute missing parameters:

Measure of Sides:

Since the side ratios are 1:1:√2, the sides can be calculated as follows:

If the length of one of the shorter sides is a, the other shorter side is also b = a
The hypotenuse (side opposite the right angle) is c = a√2

Hypotenuse:

For a 45-45-90 triangle, the hypotenuse is calculated as:

\(c = a\sqrt{2}\)

Area of 45-45-90 Triangle:

The area formula for this special right triangle is:

\(Area = \dfrac{a^2}{2}\)

You can also calculate areas of sectors and semicircles using the area of a sector calculator or the area of a semicircle calculator.

Perimeter of 45-45-90 Triangle:

\(P = 2a + c = 2b + c\)

45-45-90 Triangle Rules:

A 45-45-90 triangle is an isosceles right triangle (two sides are equal).
The hypotenuse to the sum of the legs ratio is smaller compared to other right triangles.
The altitude from the hypotenuse is maximum relative to the sum of the legs.

How To Solve a 45-45-90 Triangle?

Let’s solve a 45-45-90 triangle with a shorter side of 5 cm.

Solution:

Length of side \(a = 5 \text{ cm}\) (given)
Length of side \(b = 5 \text{ cm}\) (equal side)
Length of hypotenuse \(c = a\sqrt{2} = 5 \times 1.414 = 7.07 \text{ cm}\)

Area:

\(Area = \dfrac{a^2}{2} = \dfrac{5^2}{2} = \dfrac{25}{2} = 12.5 \text{ cm}^2\)

Perimeter:

\(P = 2a + c = 2 \times 5 + 7.07 = 14.07 \text{ cm}\)

The 45-45-90 triangle calculator provides the same results instantly, saving time on manual calculations.

Working of 45-45-90 Triangle Calculator:

Using our 45-45-90 triangle side calculator is simple and quick! The tool calculates all missing parameters based on just a few inputs.

Inputs:

Select the known parameter (side, hypotenuse, or angle).
Enter its value and choose the corresponding unit.
Click the Calculate button to get the results.

Output:

All triangle side lengths
Height measurements
Incircle and circumcircle radii
Area and perimeter

We also provide a 30-60-90 triangle calculator to calculate missing sides, angles, and other elements for that special right triangle.

FAQs:

Are All the Sides of a 45-45-90 Triangle Congruent?

No. Only the two shorter sides of a 45-45-90 triangle are congruent because it is an isosceles right triangle. The hypotenuse is longer and differs from the legs.

References:

From Wikipedia: 45-45-90 triangles, Special right triangle – angle-based, side-based, and almost-isosceles Pythagorean triples.