Right Triangle Calculator | BestCalculator

Right Triangle Calculator

Please enter a valid number
Please enter a valid number
Leg (a): -
Leg (b): -
Hypotenuse (c): -
Angle α: -
Angle β: -
Area: -
Perimeter: -
Show Calculation Steps
Formulas:
• c = √(a² + b²)
• Area = (a × b) / 2
• Perimeter = a + b + c
• Angle α = arctan(a/b)
• Angle β = 90° - α
Example:
For a=3, b=4:
c=5, Area=6, Perimeter=12, α≈36.87°, β≈53.13°
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What Is a Right Triangle?

A right triangle is a triangle that contains one angle measuring exactly 90 degrees. These triangles are central to geometry and trigonometry because of the consistent relationships between their sides and angles.

In a right triangle:

  • The hypotenuse is the longest side, located opposite the right angle.

  • The other two sides, often labeled a and b, are called the legs of the triangle.

  • The angles are commonly labeled A, B, and C, with C being the right angle (90°). In this calculator, we also use α (alpha) and β (beta) to represent the unknown angles.

The altitude (h) is the perpendicular distance from the right angle to the hypotenuse. It splits the triangle into two smaller triangles, each of which is similar to the original.

Pythagorean Triples and Right Triangles

If all three sides of a right triangle are whole numbers, the triangle is known as a Pythagorean triangle. The side lengths form a Pythagorean triple, such as:

  • 3, 4, 5

  • 5, 12, 13

  • 8, 15, 17

These specific combinations are helpful for solving many geometry problems quickly.

How to Find the Area and Perimeter

  • Perimeter (P):
    Add up all three sides:
    P = a + b + c

  • Area (A):
    Use one of these formulas:

    • A = ½ × a × b

    • A = ½ × c × h (if the altitude is known)

Special Right Triangles

30°-60°-90° Triangle

This special right triangle features angles of 30°, 60°, and 90°. The side lengths always follow the ratio:

1 : √3 : 2

This means:

  • Side opposite 30° = 1

  • Side opposite 60° = √3

  • Hypotenuse = 2

Example:
If the side opposite 60° is 5:

  • Side opposite 30° (a) = 5 / √3

  • Hypotenuse (c) = (5 × 2) / √3

This triangle is often used to evaluate trigonometric functions involving π/6.

45°-45°-90° Triangle

Also known as an isosceles right triangle, this triangle has two equal 45° angles and one 90° angle. Its side ratio is:

1 : 1 : √2

That means the two legs are the same length, and the hypotenuse is √2 times the length of either leg.

Example:
If the hypotenuse (c) is 5:

  • Each leg (a) = 5 / √2

This triangle is especially useful for solving problems involving angles like π/4.

Mathematics Calculators