Right Triangle Calculator

Published on: January 12, 2025
Final Answer: Free • Full Steps: Plus

This Right Triangle Calculator helps you find a missing side in a right triangle using two known values. It can use two side lengths to find the hypotenuse with Pythagoras’ theorem, or use an angle and one side with a trigonometric ratio to find another side. Choose the mode that matches your problem, enter the given values, and the calculator will work out the missing side. It is a simple way to check answers, understand right triangle formulas, and practise basic trigonometry step by step.

Step-by-step method

  1. Identify what is given.
  2. Write the correct formula.
  3. Substitute the values and calculate.

Formulas:

\(c = \sqrt{a^{2} + b^{2}}\)
\(\text{opposite} = \text{adjacent} \times \tan(\theta)\)

Example 1:

\(\text{Side \& Side: } a = 3,\; b = 4\)

Step 1 - Identify what is given.

In this problem: The given sides are \(a = 3\) and \(b = 4\).

\(a = 3,\; b = 4\)

Step 2 - Write the correct formula.

In this problem: Use the hypotenuse formula: \(c = \sqrt{a^{2} + b^{2}}\).

\(c = \sqrt{a^{2} + b^{2}}\)

Step 3 - Substitute the values and calculate.

In this problem: Substitute: \(c = \sqrt{3^{2} + 4^{2}} = 5\).

\(c = \sqrt{3^{2} + 4^{2}} = 5\)

Final answer:

\(c = 5\)

Example 2:

\(\text{Angle \& Side: } \theta = 30^{\circ},\; \text{adjacent} = 5\)

Step 1 - Identify what is given.

In this problem: The given values are \(\theta = 30^{\circ}\) and \(\text{adjacent} = 5\).

\(\theta = 30^{\circ},\; \text{adjacent} = 5\)

Step 2 - Write the correct formula.

In this problem: Use the tangent relationship: \(\text{opposite} = \text{adjacent} \times \tan(\theta)\).

\(\text{opposite} = \text{adjacent} \times \tan(\theta)\)

Step 3 - Substitute the values and calculate.

In this problem: Substitute: \(\text{opposite} = 5 \times \tan(30^{\circ}) \approx 2.89\).

\(\text{opposite} = 5 \times \tan(30^{\circ}) \approx 2.89\)

Final answer:

\(\text{opposite} \approx 2.89\)
See Example 1 Hide Example 1

Problem

\(\text{Side \& Side: } a = 3,\; b = 4\)

Approach

Step-by-step method

  1. Identify what is given.
  2. Write the correct formula.
  3. Substitute the values and calculate.

Step 1

Step 1 - Identify what is given.

In this problem: The given sides are \(a = 3\) and \(b = 4\).

\(a = 3,\; b = 4\)

Step 2

Step 2 - Write the correct formula.

In this problem: Use the hypotenuse formula: \(c = \sqrt{a^{2} + b^{2}}\).

\(c = \sqrt{a^{2} + b^{2}}\)

Step 3

Step 3 - Substitute the values and calculate.

In this problem: Substitute: \(c = \sqrt{3^{2} + 4^{2}} = 5\).

\(c = \sqrt{3^{2} + 4^{2}} = 5\)

Final Answer

\(c = 5\)
See Example 2 Hide Example 2

Problem

\(\text{Angle \& Side: } \theta = 30^{\circ},\; \text{adjacent} = 5\)

Approach

Step-by-step method

  1. Identify what is given.
  2. Write the correct formula.
  3. Substitute the values and calculate.

Step 1

Step 1 - Identify what is given.

In this problem: The given values are \(\theta = 30^{\circ}\) and \(\text{adjacent} = 5\).

\(\theta = 30^{\circ},\; \text{adjacent} = 5\)

Step 2

Step 2 - Write the correct formula.

In this problem: Use the tangent relationship: \(\text{opposite} = \text{adjacent} \times \tan(\theta)\).

\(\text{opposite} = \text{adjacent} \times \tan(\theta)\)

Step 3

Step 3 - Substitute the values and calculate.

In this problem: Substitute: \(\text{opposite} = 5 \times \tan(30^{\circ}) \approx 2.89\).

\(\text{opposite} = 5 \times \tan(30^{\circ}) \approx 2.89\)

Final Answer

\(\text{Angle \& Side: } \theta = 30^{\circ},\; \text{adjacent} = 5 = \text{opposite} \approx 2.89\)