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Isosceles Right Triangle Calculator

Compute hypotenuse, area, and perimeter of a 45-45-90 triangle from any known value — leg, hypotenuse, area, or perimeter.

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Understanding the Isosceles Right Triangle

An isosceles right triangle — also called a 45-45-90 triangle — is a special right triangle with two equal legs and one hypotenuse. Its two acute angles each measure exactly 45°, making it the only right triangle that is simultaneously isosceles. This geometry appears throughout mathematics, architecture, engineering, and design.

Core Variables

Leg (a): Either of the two equal shorter sides that form the right angle. Both legs are always equal in length.
Hypotenuse (c): The longest side, directly opposite the 90° angle.
Area (A): The total surface enclosed within the triangle, expressed in square units.
Perimeter (P): The total distance around the triangle, equal to the sum of all three sides.

Hypotenuse Formula Derivation

Starting from the Pythagorean theorem (c² = a² + b²) and substituting b = a, since both legs are equal in an isosceles right triangle:

c² = a² + a² = 2a²

Taking the positive square root of both sides yields: c = a√2 ≈ 1.41421 × a

According to Wolfram MathWorld's Isosceles Right Triangle reference, the √2 ratio is a fundamental constant that links leg length to hypotenuse length in every 45-45-90 triangle, regardless of the triangle's absolute size. Conversely, if the hypotenuse is known, the leg equals c / √2 = c√2 / 2.

Area Formula

The area of any triangle equals ½ × base × height. For an isosceles right triangle, the two perpendicular legs serve directly as the base and height, so no separate altitude measurement is required:

A = ½ × a × a = a² / 2

As demonstrated in Khan Academy's lesson on using the Pythagorean theorem to find triangle area, knowing only the leg length fully determines the area. If only the hypotenuse is known, substitute a = c / √2 to obtain A = c² / 4.

Perimeter Formula

Summing all three sides — two equal legs plus the hypotenuse — and factoring gives:

P = a + a + a√2 = 2a + a√2 = a(2 + √2) ≈ 3.41421 × a

To recover the leg from a known perimeter, divide: a = P / (2 + √2).

Reverse Calculations: Solving for the Leg

When a measurement other than the leg is known, rearrange the primary formulas to isolate a:

From hypotenuse: a = c / √2 ≈ 0.70711 × c
From area: a = √(2A)
From perimeter: a = P / (2 + √2) ≈ P / 3.41421

Worked Examples

Example 1 — Leg = 5 cm

Hypotenuse: c = 5√2 ≈ 7.071 cm
Area: A = 5² / 2 = 12.5 cm²
Perimeter: P = 5(2 + √2) ≈ 17.071 cm

Example 2 — Hypotenuse = 10 m

Leg: a = 10 / √2 ≈ 7.071 m
Area: A = (7.071)² / 2 ≈ 25.0 m²
Perimeter: P = 7.071 × (2 + √2) ≈ 24.142 m

Example 3 — Area = 50 ft²

Leg: a = √(2 × 50) = √100 = 10 ft
Hypotenuse: c = 10√2 ≈ 14.142 ft
Perimeter: P = 10(2 + √2) ≈ 34.142 ft

Real-World Applications

The isosceles right triangle is ubiquitous in practical settings. Roof trusses built to a 45° pitch, diagonal floor tile cuts, stair stringers, and the standard drafter's set square all follow 45-45-90 geometry. In surveying, a 45° diagonal across a square plot creates two isosceles right triangles, enabling quick diagonal distance calculations from the side length alone. Computer graphics and game engines use the 45-45-90 ratio to position objects on isometric grids. The mathematical elegance and simplicity of the √2 constant make such applications both precise and practical.

Key Properties Summary

Hypotenuse is always √2 ≈ 1.41421 times the leg length.
Interior angles sum to 180°: 45° + 45° + 90°.
The altitude from the right-angle vertex to the hypotenuse equals a / √2 = a√2 / 2.
It is the unique triangle that is simultaneously a right triangle and an isosceles triangle.

Reference

Frequently asked questions

What is an isosceles right triangle?

An isosceles right triangle is a triangle with one 90° angle and two 45° angles, where both legs — the sides forming the right angle — are equal in length. Because the isosceles and right-angle properties both hold simultaneously, it is also called the 45-45-90 triangle. The hypotenuse is always √2, approximately 1.41421, times longer than each leg, a ratio that remains constant regardless of the triangle's size.

How do you find the hypotenuse of an isosceles right triangle?

Multiply the leg length by √2, approximately 1.41421. For example, if each leg measures 8 cm, the hypotenuse equals 8 × √2 ≈ 11.314 cm. This formula derives directly from the Pythagorean theorem: c² = a² + a² = 2a², so c = a√2. When the hypotenuse is known instead and the leg is needed, divide the hypotenuse by √2, or equivalently multiply it by √2 / 2 ≈ 0.70711.

How is the area of an isosceles right triangle calculated?

Divide the square of one leg by 2: A = a² / 2. Since the two equal legs meet at a right angle, one serves as the base and the other as the height, simplifying the standard area formula ½ × base × height. For a leg of 6 m, the area is 6² / 2 = 18 m². If only the hypotenuse c is known, substitute a = c / √2 to get A = c² / 4. For example, a hypotenuse of 10 m gives an area of 100 / 4 = 25 m².

What is the perimeter formula for an isosceles right triangle?

The perimeter equals a(2 + √2), where a is the leg length. This comes from summing the two equal legs (2a) and the hypotenuse (a√2), then factoring out a. Since 2 + √2 ≈ 3.41421, a leg of 10 inches gives a perimeter of approximately 34.142 inches. To reverse-solve for the leg when only the perimeter P is known, divide: a = P / (2 + √2) ≈ P / 3.41421.

What are the angles in an isosceles right triangle?

An isosceles right triangle always contains exactly one 90° angle and two 45° angles. The two 45° angles are equal because the two legs opposite them are equal in length, satisfying the isosceles condition. All three interior angles sum to 180°, confirming the standard triangle angle-sum theorem: 45 + 45 + 90 = 180. No other combination of angles satisfies both the right-triangle and isosceles conditions simultaneously.

Where are isosceles right triangles used in real life?

Isosceles right triangles appear widely in construction, design, and navigation. Roof trusses with a 45° pitch, diagonal floor tile cuts, and stair stringers all follow 45-45-90 geometry. The standard drafter's set square is shaped as an isosceles right triangle. In surveying, a 45° diagonal across a square plot creates two isosceles right triangles, enabling quick calculation of diagonal distance from the known side length. Computer graphics use the ratio for isometric grid layouts.