Right Triangle Notes

Published on: September 29, 2026

This Right Triangle Calculator finds a missing side of a right triangle from two values that are already known, and shows the working line by line. Two legs point to the square-root relationship between the sides, while an angle and the side beside it point to the tangent instead, so what is given decides which formula is written down. The rest is substituting and working through the arithmetic. Following the same pattern every time keeps the two cases apart.

Step-by-step method

  1. Set up the problem.
  2. Write the formula for the pair that is given.
  3. Substitute the given values into the formula.
  4. Work out the value inside the formula.
  5. Calculate the missing value.

Formulas:

\(c = \sqrt{a^{2} + b^{2}}\)
\(\text{opposite} = \text{adjacent} \times \tan(\theta)\)
See Example 1 Hide Example 1

Problem

\(a = 6,\; b = 8\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula for the pair that is given.
  3. Substitute the given values into the formula.
  4. Work out the value inside the formula.
  5. Calculate the missing value.

Step 1

Step 1 - Set up the problem.

In this problem: The two legs of the right triangle are \(a = 6\) and \(b = 8\), and the hypotenuse \(c\) is missing.

\(a = 6,\; b = 8\)

Step 2

Step 2 - Write the formula for the pair that is given.

In this problem: Two legs are given, so the hypotenuse comes from \(c = \sqrt{a^{2} + b^{2}}\).

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

Step 3

Step 3 - Substitute the given values into the formula.

In this problem: Replacing \(a\) with \(6\) and \(b\) with \(8\) gives \(c = \sqrt{6^{2} + 8^{2}}\).

\(c = \sqrt{6^{2} + 8^{2}}\)

Step 4

Step 4 - Work out the value inside the formula.

In this problem: Squaring each leg and adding gives \(6^{2} + 8^{2} = 100\).

\(c = \sqrt{100}\)

Step 5

Step 5 - Calculate the missing value.

In this problem: Taking the square root of \(100\) gives \(10\).

\(c = \sqrt{100} = 10\)

Final Answer

\(c = 10\)
See Example 2 Hide Example 2

Problem

\(\theta = 45^{\circ},\; \text{adjacent} = 10\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula for the pair that is given.
  3. Substitute the given values into the formula.
  4. Work out the value inside the formula.
  5. Calculate the missing value.

Step 1

Step 1 - Set up the problem.

In this problem: The angle and the side beside it are \(\theta = 45^{\circ}\) and \(\text{adjacent} = 10\), and the opposite side is missing.

\(\theta = 45^{\circ},\; \text{adjacent} = 10\)

Step 2

Step 2 - Write the formula for the pair that is given.

In this problem: An angle and its adjacent side are given, so the opposite side comes from \(\text{opposite} = \text{adjacent} \times \tan(\theta)\).

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

Step 3

Step 3 - Substitute the given values into the formula.

In this problem: Replacing \(\text{adjacent}\) with \(10\) and \(\theta\) with \(45^{\circ}\) gives \(\text{opposite} = 10 \times \tan(45^{\circ})\).

\(\text{opposite} = 10 \times \tan(45^{\circ})\)

Step 4

Step 4 - Work out the value inside the formula.

In this problem: The tangent of \(45^{\circ}\) is \(1\).

\(\text{opposite} = 10 \times 1\)

Step 5

Step 5 - Calculate the missing value.

In this problem: Multiplying \(10\) by \(1\) gives \(10\).

\(\text{opposite} = 10 \times 1 = 10\)

Final Answer

\(\text{opposite} = 10\)
See Example 3 Hide Example 3

Problem

\(a = 5,\; b = 12\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula for the pair that is given.
  3. Substitute the given values into the formula.
  4. Work out the value inside the formula.
  5. Calculate the missing value.

Step 1

Step 1 - Set up the problem.

In this problem: The two legs of the right triangle are \(a = 5\) and \(b = 12\), and the hypotenuse \(c\) is missing.

\(a = 5,\; b = 12\)

Step 2

Step 2 - Write the formula for the pair that is given.

In this problem: Two legs are given, so the hypotenuse comes from \(c = \sqrt{a^{2} + b^{2}}\).

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

Step 3

Step 3 - Substitute the given values into the formula.

In this problem: Replacing \(a\) with \(5\) and \(b\) with \(12\) gives \(c = \sqrt{5^{2} + 12^{2}}\).

\(c = \sqrt{5^{2} + 12^{2}}\)

Step 4

Step 4 - Work out the value inside the formula.

In this problem: Squaring each leg and adding gives \(5^{2} + 12^{2} = 169\).

\(c = \sqrt{169}\)

Step 5

Step 5 - Calculate the missing value.

In this problem: Taking the square root of \(169\) gives \(13\).

\(c = \sqrt{169} = 13\)

Final Answer

\(c = 13\)
See Example 4 Hide Example 4

Problem

\(\theta = 30^{\circ},\; \text{adjacent} = 8\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formula for the pair that is given.
  3. Substitute the given values into the formula.
  4. Work out the value inside the formula.
  5. Calculate the missing value.

Step 1

Step 1 - Set up the problem.

In this problem: The angle and the side beside it are \(\theta = 30^{\circ}\) and \(\text{adjacent} = 8\), and the opposite side is missing.

\(\theta = 30^{\circ},\; \text{adjacent} = 8\)

Step 2

Step 2 - Write the formula for the pair that is given.

In this problem: An angle and its adjacent side are given, so the opposite side comes from \(\text{opposite} = \text{adjacent} \times \tan(\theta)\).

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

Step 3

Step 3 - Substitute the given values into the formula.

In this problem: Replacing \(\text{adjacent}\) with \(8\) and \(\theta\) with \(30^{\circ}\) gives \(\text{opposite} = 8 \times \tan(30^{\circ})\).

\(\text{opposite} = 8 \times \tan(30^{\circ})\)

Step 4

Step 4 - Work out the value inside the formula.

In this problem: The tangent of \(30^{\circ}\) is \(0.58\).

\(\text{opposite} = 8 \times 0.58\)

Step 5

Step 5 - Calculate the missing value.

In this problem: Multiplying \(8\) by \(0.58\) gives \(4.62\).

\(\text{opposite} = 8 \times 0.58 \approx 4.62\)

Final Answer

\(\text{opposite} \approx 4.62\)