Mensuration: area and volume
Perimeter and area of 2D shapes, circles, arcs and sectors, and the surface area and volume of solids.
📌 Key points
- Areas: rectangle = length × width; triangle = 12 × base × height; parallelogram = base × height; trapezium = 12(a + b)h.
- Circle: circumference = 2πr = πd; area = πr2. Check whether you are given the radius or the diameter.
- Arc length = θ360 × 2πr and sector area = θ360 × πr2, where θ is the angle at the centre.
- The perimeter of a sector = arc length + 2 radii.
- Volume of a prism = area of cross-section × length; a cylinder has volume πr2h and curved surface area 2πrh.
- Pyramid volume = 13 × base area × height. Cone: V = 13πr2h, curved surface = πrl (l = slant height). Sphere: V = 43πr3, surface area = 4πr2.
- Surface area is the total area of all the faces; sketching the net helps you not to miss any.
- Area and volume units: 1 m2 = 10 000 cm2, 1 cm2 = 100 mm2, 1 m3 = 1 000 000 cm3, and 1 litre = 1000 cm3.
- For compound shapes, split them into simple shapes, then add (or subtract) their areas or volumes.
📖 Key terms
- Perimeter
- The total distance around the edge of a shape.
- Area
- The amount of flat space a shape covers, in square units.
- Volume
- The amount of space a solid takes up, in cubic units.
- Surface area
- The total area of all the faces of a solid.
- Arc
- Part of the circumference of a circle.
- Sector
- The region between two radii and an arc.
- Cross-section
- The shape you see when you cut straight through a solid.
- Slant height
- The distance from the apex of a cone to the edge of its base, along the surface.
✏️ Worked example
A sector has radius 9 cm and angle 80°. Find its perimeter and its area.
- Arc length = 80360 × 2 × π × 9 = 4π = 12.566… cm.
- Perimeter = arc + two radii = 12.566 + 9 + 9 = 30.566… cm.
- Area = 80360 × π × 92 = 18π = 56.548… cm2.
- Answer: perimeter = 30.6 cm, area = 56.5 cm2 (3 s.f.).
The perimeter of a sector or semicircle includes the straight edges as well as the curved arc — forgetting the radii is one of the most common lost marks.
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