Right Triangle Calculator

How to Calculate Right Triangle Properties

A right triangle is a triangle with one 90-degree angle. It has unique properties that make it essential in geometry and trigonometry. Here's how to calculate various properties of a right triangle:

Right Triangle Formulas

1. Pythagorean Theorem:

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

   Where a and b are the lengths of the legs, and c is the length of the hypotenuse.

2. Trigonometric Ratios:

   \( \sin A = \frac{opposite}{hypotenuse} = \frac{a}{c} \)

   \( \cos A = \frac{adjacent}{hypotenuse} = \frac{b}{c} \)

   \( \tan A = \frac{opposite}{adjacent} = \frac{a}{b} \)

   Where A is one of the non-right angles in the triangle.

3. Area:

   \( Area = \frac{1}{2} \times base \times height \)

4. Perimeter:

   \( Perimeter = a + b + c \)

Calculation Steps

1. Identify the known values (side lengths or angles).
2. If you have two side lengths, use the Pythagorean Theorem to find the third side.
3. If you have a side length and an angle, use trigonometric ratios to find other sides or angles.
4. Calculate the area using the formula: \( Area = \frac{1}{2} \times base \times height \).
5. Calculate the perimeter by adding all side lengths.
6. Use trigonometric ratios to find any remaining unknown angles.

Example Calculation

Let's solve a right triangle with leg a = 3 units and leg b = 4 units.

1. Given: a = 3 units, b = 4 units
2. Use the Pythagorean Theorem to find c:

   \( c^2 = a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25 \)

   \( c = \sqrt{25} = 5 \) units

3. Calculate angles:

   \( \sin A = \frac{a}{c} = \frac{3}{5} \)

   \( A = \arcsin(\frac{3}{5}) \approx 36.87° \)

   \( B = 90° - A \approx 53.13° \)

4. Calculate area:

   \( Area = \frac{1}{2} \times 3 \times 4 = 6 \) square units

5. Calculate perimeter:

   \( Perimeter = 3 + 4 + 5 = 12 \) units

Visual Representation

This diagram illustrates the right triangle from our example, showing how the Pythagorean Theorem and trigonometric ratios relate the sides and angles of the triangle.