0580

Volume and Surface Area

Lengths, Areas and Volumes

Prisms and cylinders

  • A prism is any solid with the same cross-section all the way along, and its volume is that cross-sectional area times its length
  • On this specification "prism" is defined broadly, so a cylinder is a prism with a circular cross-section, and even a cylindrical sector counts
  • So a triangular prism whose cross-section has area 24 cm² and whose length is 15 cm has volume 24 × 15 = 360 cm³
  • A cuboid is a prism with a rectangular cross-section, and its volume is length × width × height
A cuboid with each of its three dimensions arrowed and named: length along the front bottom edge, width running back, and height up the side
Source: Volume by Save My Exams
  • For a cylinder the cross-section is a circle, which is why V = πr²h
A cylinder with the radius r marked across the top face from its centre and the height h marked up the side, the hidden part of the base drawn as a dashed curve
Source: Volume by Save My Exams

Pyramids, cones and spheres

  • A pyramid has volume ⅓ × base area × perpendicular height, and the height must run straight down from the apex to the base
A square-based pyramid with a red arrow running straight down from the apex to the centre of the base, marked h, which is the perpendicular height rather than the length of a sloping edge
Source: Volume by Save My Exams
  • A cone is a pyramid on a circular base, so its volume is ⅓πr²h, again using the perpendicular height
  • The sloping edge of a cone is longer than its perpendicular height, and Pythagoras converts between them since the radius, the height and the sloping edge form a right-angled triangle
A cone with the perpendicular height h drawn from the apex down to the centre of the base, a small square marking the right angle where it meets the radius r
Source: Volume by Save My Exams
  • A sphere has volume 4/3 πr³, and a hemisphere is half of that
A sphere with its radius r drawn from the centre out to the surface, the equator shown as a curve with its hidden half dashed
Source: Volume by Save My Exams
Worked example

A cone from its sloping edge

A cone has base radius 5 cm and sloping edge 13 cm. Find its volume in terms of π.

A cone standing on its circular base. The sloping edge from the rim up to the apex is labelled 13 cm and the base radius is labelled 5 cm. A dashed vertical line runs from the centre of the base to the apex and a dashed radius runs out to the rim, meeting at a marked right angle; the vertical height carries no label. The figure is marked NOT TO SCALE.

Solution:

  • The volume formula needs the perpendicular height, but the sloping edge is given
  • The radius, the height and the sloping edge form a right-angled triangle, with the sloping edge as the hypotenuse
  • By Pythagoras, h² = 13² − 5² = 169 − 25 = 144
  • So h = 12 cm
  • Volume = ⅓ × π × 5² × 12 = ⅓ × π × 25 × 12
  • Volume = 100π cm³