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E5.4 Surface area and volume

Syllabus
0580–2028–2029
Topic
E5.4
Level
Extended

Learning objectives

E5.4.1Carry out calculations and solve problems involving the surface area• Carry out calculations and solve problems involving the surface area and volume of a: • cuboid • prism • cylinder • sphere • pyramid • cone. Answers may be asked for in terms of π. The following formulas are given in the List of formulas: • curved surface area of a cylinder • curved surface area of a cone • surface area of a sphere • volume of a prism • volume of a pyramid • volume of a cylinder • volume of a cone • volume of a sphere. The term prism refers to any solid with a uniform cross-section, e.g. a cylindrical sector.

Calculate surface area and volume of solids

Surface area totals the exposed two-dimensional faces of a solid; volume measures the three-dimensional space inside it. Choose the quantity first, then identify the dimensions the matching relationship needs.

Solid Total surface area Volume
cuboid l×w×hl\times w\times h 2(lw+lh+wh)2(lw+lh+wh) lwhlwh
prism, cross-section area BB, perimeter PP, length LL 2B+PL2B+PL BLBL
cylinder, radius rr, height hh 2πr2+2πrh2\pi r^2+2\pi rh πr2h\pi r^2h
sphere, radius rr 4πr24\pi r^2 43πr3\frac43\pi r^3
pyramid, base area BB, perpendicular height hh base plus every triangular face 13Bh\frac13Bh
cone, radius rr, perpendicular height hh, slant height ss πr2+πrs\pi r^2+\pi rs 13πr2h\frac13\pi r^2h

For a surface-area problem, list each exposed face or curved surface once; a closed cylinder and cone include their circular bases. For volume, a uniform prism—including a cylinder—is cross-section area multiplied by length, while a pyramid or cone has one third of the volume of a matching prism with the same base and perpendicular height. Keep every length in one unit before calculating, then use square units for surface area and cubic units for volume. For an unknown dimension, write the full equation and rearrange.

A closed cylinder has radius 33 cm and height 88 cm. Its total surface area is 2π(3)2+2π(3)(8)=66π cm22\pi(3)^2+2\pi(3)(8)=66\pi\text{ cm}^2, while its volume is π(3)2(8)=72π cm3\pi(3)^2(8)=72\pi\text{ cm}^3. Leaving π\pi exact keeps both values accurate until a decimal is requested.

A cone's slant height ss belongs in curved surface area, but its perpendicular height hh belongs in volume. Do not include internal joins in surface area. Compound solids, removed pieces and fractions of solids belong to the next Topic.