Geometry and measures - Mensuration and calculation: know the formulae for: Pythagoras’ theorem a² + b² = c² , and the trigonometric ratios, sin θ = opposite/hypotenuse , cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent ; apply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two and three dimensional figures (Higher tier)

The idea

Higher tier only. Know Pythagoras’ theorem, a² + b² = c², and the trigonometric ratios SOH CAH TOA. Apply them to find lengths and angles in right-angled triangles. For example, if the opposite side is 5 cm and the hypotenuse 13 cm, sin θ = 5/13, so θ = 22.6°. In 3D figures, identify a right-angled triangle, often using Pythagoras twice. In a cuboid 3cm by 4cm by 12cm, the base diagonal is √(3²+4²) = 5cm, and the space diagonal is √(5²+12²) = 13cm. The angle it makes with the base has tan θ = 12/5, so θ = 67.4°. For general triangles, create right-angled triangles by dropping a perpendicular. In an isosceles triangle with sides 10, 10, 12, a perpendicular to the base of 12 creates two triangles with base 6 and hypotenuse 10, so the height is √(10²-6²) = 8cm.

What earns the marks

• Stating and using Pythagoras' theorem a² + b² = c² with the hypotenuse correctly identified.
• Selecting the correct trigonometric ratio (sin, cos or tan) for the sides and angle given.
• Constructing a perpendicular in a general triangle to form right-angled triangles.
• Identifying and drawing the correct right-angled triangle within a 3D solid.
• A first correct use of Pythagoras or a trigonometric ratio to find an intermediate length.
• Calculating the final length or angle to the required degree of accuracy.

Common mistakes and how to put them right

• Applying Pythagoras' theorem or SOH CAH TOA directly to a non-right-angled triangle without first constructing a perpendicular.

• In 3D problems, using the wrong angle, such as the angle with an edge instead of the angle with a plane.

• Rounding an intermediate length and using the rounded value in the next stage of calculation, causing inaccuracies.

• Mixing up the sides, for example using adjacent/hypotenuse for tan.

Exam tips

• Sketch each triangle you use separately and flat on the page, labelling the sides and angles you know.
• If a triangle is not right-angled, see if you can drop a perpendicular height to create two that are.
• Keep full calculator accuracy for intermediate steps and only round the final answer.