22 min read

Finding Missing Sides

Use SOH CAH TOA to find a side when you know an angle and one other side.

Choosing the Right Ratio

To find a missing side: label the sides relative to the known angle (Opposite, Adjacent, Hypotenuse), choose whichever ratio connects the side you know to the side you want, then rearrange and solve.

Example 1 — finding the opposite side

A right-angled triangle has an angle of 30 ° and a hypotenuse of 10 . Find the opposite side. 1 The known side is the hypotenuse, and we want the opposite side. Use sin: sin(θ) = Opposite \div Hypotenuse 2 sin(30 °) = Opposite / 10 3 Rearrange: Opposite = 10 \times sin(30 °) = 10 \times 0.5 = 5 Answer: 5

Example 2 — finding the adjacent side

A right-angled triangle has an angle of 60 ° and a hypotenuse of 8 . Find the adjacent side. 1 We know the hypotenuse and want the adjacent side. Use cos: cos(θ) = Adjacent \div Hypotenuse 2 cos(60 °) = Adjacent / 8 3 Rearrange: Adjacent = 8 \times cos(60 °) = 8 \times 0.5 = 4 Answer: 4

Example 3 — finding a side using tan

A right-angled triangle has an angle of 40 ° and an adjacent side of 12 . Find the opposite side. 1 We know the adjacent side and want the opposite side. Use tan: tan(θ) = Opposite \div Adjacent 2 tan(40 °) = Opposite \div 12 3 Rearrange: Opposite = 12 \times tan(40 °) \approx 12 \times 0.839 \approx 10.07 Answer: \approx 10.07

Practice

A right-angled triangle has an angle of 25 ° and a hypotenuse of 20 . Find the opposite side, to 2 decimal places.

Your answer