Home Back

Isosceles Right Triangle Calculator

Isosceles Right Triangle

square

1. What is the Isosceles Right Triangle Calculator?

Definition: The Isosceles Right Triangle Calculator computes the hypotenuse, area, perimeter, and height of an isosceles right triangle, which has two equal legs and angles of 45°, 45°, and 90°.

Purpose: Assists in calculating geometric properties for isosceles right triangles used in construction, design, or education with customizable units.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Hypotenuse:

\[ h = l \cdot \sqrt{2} \]

Area:

\[ A = \frac{1}{2} \cdot l \cdot l = \frac{l^2}{2} \]

Perimeter:

\[ P = 2 \cdot l + h = 2 \cdot l + l \cdot \sqrt{2} \]

Height (to Hypotenuse):

\[ H = \frac{l}{\sqrt{2}} \]
Where:
  • \( l \): Length of each leg (equal sides).
  • \( h \): Hypotenuse.
  • \( A \): Area.
  • \( P \): Perimeter.
  • \( H \): Height from the right angle to the hypotenuse.

Steps:

  • Step 1: Input Leg Length. Enter a positive value for the length of one leg.
  • Step 2: Select Unit. Choose from cm, m, inch, feet, or yard.
  • Step 3: Calculate. Compute the hypotenuse, area, perimeter, and height with fixed 4 decimal place precision.

3. Importance of Isosceles Right Triangle Calculation

Calculating isosceles right triangle properties is crucial for:

  • Construction: Designing right-angled structures with equal legs.
  • Design: Creating symmetrical patterns or frameworks.
  • Education: Teaching special right triangle properties and the Pythagorean theorem.

4. Using the Calculator

Example 1: Leg Length 5 cm:

  • Step 1: Leg Length = 5 cm.
  • Step 2: Hypotenuse = \( 5 \cdot \sqrt{2} \approx 7.0711 \) cm.
  • Step 3: Area = \( \frac{5^2}{2} = 12.5000 \) square cm.
  • Step 4: Perimeter = \( 2 \cdot 5 + 7.0711 \approx 17.0711 \) cm.
  • Step 5: Height = \( \frac{5}{\sqrt{2}} \approx 3.5355 \) cm.
  • Result: Hypotenuse ≈ 7.0711 cm, Area = 12.5000 square cm, Perimeter ≈ 17.0711 cm, Height ≈ 3.5355 cm.

Example 2: Leg Length 3 inches:

  • Step 1: Leg Length = 3 inches.
  • Step 2: Hypotenuse = \( 3 \cdot \sqrt{2} \approx 4.2426 \) inches.
  • Step 3: Area = \( \frac{3^2}{2} = 4.5000 \) square inches.
  • Step 4: Perimeter = \( 2 \cdot 3 + 4.2426 \approx 10.2426 \) inches.
  • Step 5: Height = \( \frac{3}{\sqrt{2}} \approx 2.1213 \) inches.
  • Result: Hypotenuse ≈ 4.2426 inches, Area = 4.5000 square inches, Perimeter ≈ 10.2426 inches, Height ≈ 2.1213 inches.

Example 3: Leg Length 2 m:

  • Step 1: Leg Length = 2 m.
  • Step 2: Hypotenuse = \( 2 \cdot \sqrt{2} \approx 2.8284 \) m.
  • Step 3: Area = \( \frac{2^2}{2} = 2.0000 \) square m.
  • Step 4: Perimeter = \( 2 \cdot 2 + 2.8284 \approx 6.8284 \) m.
  • Step 5: Height = \( \frac{2}{\sqrt{2}} \approx 1.4142 \) m.
  • Result: Hypotenuse ≈ 2.8284 m, Area = 2.0000 square m, Perimeter ≈ 6.8284 m, Height ≈ 1.4142 m.

Example 4: Leg Length 4 feet:

  • Step 1: Leg Length = 4 feet.
  • Step 2: Hypotenuse = \( 4 \cdot \sqrt{2} \approx 5.6569 \) feet.
  • Step 3: Area = \( \frac{4^2}{2} = 8.0000 \) square feet.
  • Step 4: Perimeter = \( 2 \cdot 4 + 5.6569 \approx 13.6569 \) feet.
  • Step 5: Height = \( \frac{4}{\sqrt{2}} \approx 2.8284 \) feet.
  • Result: Hypotenuse ≈ 5.6569 feet, Area = 8.0000 square feet, Perimeter ≈ 13.6569 feet, Height ≈ 2.8284 feet.

5. Frequently Asked Questions (FAQ)

Q: What is an isosceles right triangle?
A: An isosceles right triangle is a right-angled triangle with two equal legs and angles of 45°, 45°, and 90°.

Q: How is the height calculated?
A: The height from the right angle to the hypotenuse is \( \frac{l}{\sqrt{2}} \), derived from the triangle's symmetry.

Q: Can the calculator handle other units?
A: Yes, it supports cm, m, inch, feet, and yard with automatic conversion.

Isosceles Right Triangle Calculator© - All Rights Reserved 2025