Isosceles Right Triangle Area: Formula & Calculator

This calculator works out the area of an isosceles right triangle from the length of one leg (the two equal sides). Enter the leg length, and you get the area in square units, along with the hypotenuse length and the triangle’s height. The formula is Area = (leg × leg) / 2, which is half the square built on the leg.

Formula

For an isosceles right triangle, the two legs (the sides that form the right angle) are equal in length. Let that length be a. The area A is:

A = (a × a) / 2 = a² / 2

Where:

• a = length of each leg (same unit as the area’s square unit)
• A = area (square units, e.g., m², cm², ft²)

The hypotenuse (the side opposite the right angle) is a√2, by the Pythagorean theorem. The height measured perpendicular to the hypotenuse is a/√2, which is also the radius of the circumscribed circle.

Worked Example

Using the default leg length of 5 m:

1. Square the leg: 5 × 5 = 25 m²
2. Divide by 2: 25 / 2 = 12.5 m²
3. Hypotenuse: 5 × √2 ≈ 5 × 1.41421 = 7.07 m (rounded to 2 decimals)
4. Height to hypotenuse: 5 / √2 ≈ 5 / 1.41421 = 3.54 m (rounded to 2 decimals)

The calculator shows an area of 12.50 m², hypotenuse of 7.07 m, and height of 3.54 m.

Why This Formula Works

Any right triangle’s area is half the product of its two legs (base × height / 2). In an isosceles right triangle, both legs are the same, so you multiply the leg by itself and halve it. This is the same as taking half of the area of a square with side length a – cut a square along its diagonal, and you get two identical isosceles right triangles.

This is also a special case of the 45-45-90 triangle, where the ratio of sides is 1 : 1 : √2. The area formula is simpler than the general triangle area formula because you don’t need a separate base and height – the legs serve both roles.

When to Use This Calculator

Use this when you know the length of one of the equal sides (legs) and need the area. Common situations include:

• Finding the area of a right-angled roof truss or gable end
• Calculating the size of a triangular plot where two sides meet at 90° and are equal
• Working out the area of a folded paper or fabric cut along a diagonal
• Solving geometry homework problems that give the leg length

If you instead know the hypotenuse, you can still use this calculator by first dividing the hypotenuse by √2 to get the leg length, then entering that value.

Common Mistakes and How to Avoid Them

• Using the hypotenuse as a leg: If you enter the hypotenuse (the longest side) as the leg, the area will be double the correct value. The hypotenuse is always the side opposite the right angle.
• Unit mix-ups: If the leg is in meters, the area is in square meters. If you measure in feet, the area is in square feet. Convert all measurements to the same unit before calculating.
• Forgetting to halve: The most common error is squaring the leg and stopping there. That gives the area of a square, not the triangle. Always divide by 2.
• Rounding too early: When using the hypotenuse method, round only at the final step. For example, if hypotenuse = 10, leg = 10/√2 ≈ 7.071, not 7.07. Using 7.07 gives an area of 24.99 instead of 25.00.

Limitations and Alternatives

This formula applies only to right triangles where the two legs are equal. For other right triangles (e.g., 3-4-5), the area is (base × height)/2 with different base and height. For non-right triangles, use Heron’s formula or (1/2)ab·sin(C).

The formula also assumes Euclidean geometry. On a curved surface (like the Earth’s surface for large areas), the actual area differs, but for everyday sizes, the difference is negligible.

If you know only the perimeter or the hypotenuse, you can derive the leg length first:

• From hypotenuse c: leg a = c / √2
• From perimeter P: leg a = P / (2 + √2)

Frequently Asked Questions

What is the area of an isosceles right triangle with legs of 1?

The area is 1 × 1 / 2 = 0.5 square units. This is the smallest integer leg length that gives a non-integer area, which often surprises students.

How do I find the area if I only know the hypotenuse?

Divide the hypotenuse by √2 to get the leg length, then square and halve. For example, hypotenuse 10 gives leg ≈ 7.071, area ≈ 25.00. You can also use the direct formula: area = c² / 4.

Is an isosceles right triangle always a 45-45-90 triangle?

Yes. Because the two legs are equal, the two acute angles must be equal, and since they sum to 90°, each is 45°. So every isosceles right triangle has angles 45°, 45°, and 90°.

Can I use this formula for any right triangle?

No. Only for right triangles where the legs are equal. For a general right triangle, use (base × height)/2 with the two legs as base and height, but they must be different values.

What if my leg length is in inches and I want square feet?

Convert the leg to feet first (divide by 12), then use the formula. For example, a 24-inch leg is 2 feet, giving area = 2²/2 = 2 ft². Alternatively, calculate in square inches and divide by 144.

Why is the height to the hypotenuse not equal to the leg?

The height to the hypotenuse is the perpendicular distance from the right angle to the hypotenuse. It is always shorter than the leg. In a 45-45-90 triangle, that height equals half the hypotenuse, or a/√2, which is about 0.707 times the leg.

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in a | Leg length | m | 5 | 0
out area | Area | m² | (a * a) / 2 | 2
out hyp | Hypotenuse | m | a * sqrt(2) | 2
out height | Height to hypotenuse | m | a / sqrt(2) | 2