Mensuration and calculation: apply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two and three dimensional figures (Higher tier only)

The idea

This Higher tier row applies Pythagoras’ theorem and the trigonometric ratios to right-angled triangles in two and three dimensions. In a solid, identify a right-angled triangle within a face or use a face or space diagonal, then apply a² + b² = c² or sin, cos or tan. For a general triangle, use the given information to identify or construct a right-angled triangle where possible; the sine and cosine rules belong to the separate Higher-tier criterion on those rules. Always identify the relevant triangle before choosing a method.

What earns the marks

• Apply Pythagoras’ theorem to find a length in a right-angled triangle within a two-dimensional or three-dimensional figure.
• Apply trigonometric ratios to find an angle or length in a right-angled triangle within a two-dimensional or three-dimensional figure.
• Identify or construct a suitable right-angled triangle within a three-dimensional figure, such as a face diagonal or space diagonal, and use it to find the required length or angle.
• For a general triangle, recognise when a right-angled decomposition is possible and state what additional information or construction is needed when it is not.

Common mistakes and how to put them right

• Assuming every triangle is right-angled.

  Put it right by checking for a right angle and using a suitable construction or stated method when no right angle is present.

• Failing to identify the correct right-angled triangle in a three-dimensional figure.

  Put it right by sketching the relevant face or cross-section and labelling the known lengths.

• Using a sloping edge or diagonal without checking which triangle it belongs to; correct by drawing the relevant face or cross-section before calculating.

• Rounding too early in a multi-step calculation.

  Put it right by keeping full calculator precision until the final answer.

Exam tips

• Draw a clear diagram of the two-dimensional or three-dimensional figure and mark all known lengths and angles.
• For three-dimensional problems, extract and redraw the relevant right-angled triangle before calculating.
• Check that the selected triangle is genuinely right-angled, label its sides relative to the angle, and avoid rounding until the final answer.