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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 c:

Height

cm ▾

trianle details image

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The 45-45-90 triangle calculator helps you compute all sides, area, perimeter, inradius, and circumradius for this special right triangle. Simply input any known parameter, and the tool calculates all other values instantly and accurately.

What Is a 45-45-90 Triangle?

A 45-45-90 triangle is a special type of right triangle where two 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)

Formulas for 45-45-90 Triangle

The calculator uses trigonometry-based formulas to determine all missing parameters.

Sides

With side ratios 1:1:√2:

If a shorter side is a, the other short side is b = a
The hypotenuse is c = a√2

Hypotenuse

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

Area

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

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

Perimeter

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

Key Rules for 45-45-90 Triangles

It is an isosceles right triangle, with two equal sides.
The hypotenuse is longer than either leg.
The altitude from the hypotenuse is the maximum relative to the legs.

Example: Solving a 45-45-90 Triangle

Given a shorter side \(a = 5 \text{ cm}\):

Other short side \(b = 5 \text{ cm}\)
Hypotenuse \(c = a\sqrt{2} = 5 \times 1.414 = 7.07 \text{ cm}\)

Area:

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

Perimeter:

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

The 45-45-90 triangle calculator provides all these results instantly, saving time and effort.

Using the 45-45-90 Triangle Calculator

Input:

Enter the known parameter (side, hypotenuse, or angle).
Choose the unit (cm, m, inch, etc.).
Click Calculate to get all results.

Output:

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

You may also use our 30-60-90 triangle calculator for solving that special right triangle.

FAQs

Are all sides of a 45-45-90 triangle equal?

No. Only the two legs are equal. The hypotenuse is longer and differs from the legs.

References

From Wikipedia: 45-45-90 triangles, Special right triangles – angle ratios, side ratios, and Pythagorean triples.

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