SOHCAHTOA Finding Missing Sides –

Finding a missing side using SOHCAHTOA is one of the most important trigonometry skills in GCSE Maths. You need one angle and one side of a right-angled triangle, and the correct trig ratio does the rest.

What Is SOHCAHTOA for Finding Sides?

SOHCAHTOA is a mnemonic that links the three trigonometric ratios — sine, cosine and tangent — to the sides of a right-angled triangle. When you know an angle (other than the right angle) and one side, you can use the appropriate ratio to calculate any other side.

The key is labelling the sides correctly relative to the angle you are working with. The Opposite (O) is across from the angle, the Adjacent (A) is next to the angle (but not the hypotenuse), and the Hypotenuse (H) is always opposite the right angle and always the longest side.

Once the sides are labelled, you choose the ratio that connects the known side to the unknown side, substitute, rearrange, and calculate. Your calculator must be in degree mode for GCSE questions.

Key Formulas

Step-by-Step Method

1. Label the sides O, A, and H relative to the given angle.
2. Identify the known side and the unknown side.
3. Choose the ratio that uses both (SOH, CAH, or TOA).
4. Write the ratio as an equation and substitute.
5. Rearrange to make the unknown side the subject.
6. Calculate using your calculator in degree mode.

Worked Example 1 — Foundation Level

Question: In a right-angled triangle, the angle is 40° and the hypotenuse is 12 cm. Find the side opposite the 40° angle. Give your answer to 1 decimal place.

Working: Label: H = 12, O = ?, angle = 40°. O and H are involved, so use sin (SOH). sin 40° = O ÷ 12 O = 12 × sin 40° O = 12 × 0.6428

Answer: O = 7.7 cm (1 d.p.).

Worked Example 2 — Higher Level

Question: In a right-angled triangle, the angle is 62° and the side opposite is 15 cm. Find the hypotenuse. Give your answer to 3 significant figures.

Working: Label: O = 15, H = ?, angle = 62°. O and H are involved, so use sin (SOH). sin 62° = 15 ÷ H H = 15 ÷ sin 62° H = 15 ÷ 0.8829

Answer: H = 17.0 cm (3 s.f.).

Worked Example 3 — Exam Style

Question: A ramp makes an angle of 25° with the horizontal ground. The ramp is 8 m long. How high does the ramp rise vertically? Give your answer to 1 decimal place.

Working: The ramp is the hypotenuse (H = 8 m). The vertical rise is opposite the 25° angle (O = ?). sin 25° = O ÷ 8 O = 8 × sin 25° O = 8 × 0.4226

Answer: The ramp rises 3.4 m (1 d.p.).

Common Mistakes

• Mislabelling O and A. The opposite and adjacent depend entirely on which angle you are working with. If you switch to a different angle, the labels swap.
• Multiplying when you should divide (or vice versa). If the unknown is on the bottom of the fraction, you need to divide. If it is on the top, you multiply. Use a formula triangle if this helps.
• Calculator in radian mode. GCSE questions use degrees. Check your calculator display shows D or DEG, not R or RAD. An answer like 0.12 instead of 7.7 is a sign your calculator is in radians.

Exam Tips

• Write the ratio name (e.g. "sin 40° = O/H") before substituting — this earns a method mark even if your final answer has a rounding error.
• If the unknown side is the hypotenuse, you will always divide by the trig value.
• If the unknown side is O or A, you will always multiply.
• Show every line of working: label, ratio, substitution, calculation, answer.

Practice Questions

Q1 (Foundation): The angle is 55° and the adjacent side is 10 cm. Find the opposite side to 1 d.p.

Q2 (Foundation): The angle is 30° and the hypotenuse is 20 cm. Find the adjacent side.

Q3 (Higher): The angle is 48° and the opposite side is 9 cm. Find the adjacent side to 1 d.p.

Practise SOHCAHTOA finding sides questions with instant feedback — completely free on GCSEMathsAI.

Related Topics

• Trigonometry SOHCAHTOA
• SOHCAHTOA Finding Missing Angles
• Pythagoras' Theorem

Summary

• Label the sides O, A, and H relative to the given angle before choosing a ratio.
• Use SOH when O and H are involved, CAH for A and H, and TOA for O and A.
• If the unknown is on top of the fraction, multiply; if on the bottom, divide.
• Always check your calculator is in degree mode and show every step of working for full method marks.