Geometry and measures - Mensuration and calculation: know and apply formulae to calculate: area of triangles, parallelograms, trapezia; volume of cuboids and other right prisms (including cylinders)

The idea

The area of a triangle is half the base times the perpendicular height, the area of a parallelogram is base times perpendicular height, and the area of a trapezium is ½ × (a + b) × h, where a and b are the two parallel sides and h is the gap between them. The height has to be perpendicular to the base, never the slanted edge. A prism has the same cross-section all the way along, so the volume of a right prism is the area of that cross-section multiplied by its length, and for a cylinder the cross-section is a circle, which gives volume = πr²h. A trapezium with parallel sides of 5 cm and 9 cm and a height of 4 cm has area ½ × 14 × 4 = 28 cm². A cylinder of radius 3 cm and height 10 cm has volume 90π cm³, which is about 282.7 cm³. A cuboid is a right prism whose cross-section is a rectangle, so its volume is length × width × height.

What earns the marks

• A correct substitution into the right formula, such as ½ × (5 + 9) × 4 for a trapezium, even if the arithmetic then goes wrong.
• Using a perpendicular height rather than a slant length taken off the diagram.
• The area of the cross-section of a prism found before it is multiplied by the length.
• The final value with the correct unit, square units for an area and cube units for a volume.
• Recognising a cuboid as a right prism and using length × width × height, or cross-section area × length.

Common mistakes and how to put them right

• Using the sloping side of a triangle or parallelogram as its height.

• Leaving out the half in the triangle and trapezium formulae, so the area comes out doubled.

• Multiplying the two parallel sides of a trapezium instead of adding them, or using only one of them.

• Squaring the diameter of a cylinder in place of the radius, which makes the volume four times too big.

Exam tips

• Label the base and the perpendicular height on the diagram before you substitute anything.
• For a prism, find the cross-section area first and write it on its own line; this keeps the method clear and makes it easier to check.
• Leave an answer in terms of π only when the question asks for that; otherwise round as instructed and add the unit.