CAIE IGCSE Mathematics Extended Solid Mensuration

Practise multi-step surface-area and volume problems involving linked dimensions, composite solids, conservation of material and equal-measure equations.

Exam points

calculate a missing perpendicular height with Pythagoras before applying a solid formula
equate volumes or surface areas when material is recast or two solid measures are stated equal
combine unit conversion, flow or capacity information with a cuboid or curved-solid volume

Covered syllabus points

E5.4.1 Carry out calculations and solve problems involving the surface area

(a)

A cylinder with radius 6 cm and height $h \mathrm{~cm}$ has the same volume as a sphere with radius 4.5 cm. Find the value of h.
[The volume, V, of a sphere with radius r is $V=\frac{4}{3} \pi r^{3}$ .]

3.375 cao
M2 for $\frac{\frac{4}{3}\pi(4.5)^3}{\pi6^2}$ , or M1 for a correct volume equation.

(b)

A solid metal cube of side 20 cm is melted down and made into 40 solid spheres, each of radius $r \mathrm{~cm}$ .

Find the value of r.
[The volume, V, of a sphere with radius r is $V=\frac{4}{3} \pi r^{3}$ .]

3.63 or 3.627 to 3.628
M2 for $\frac{20^3}{40\times\frac{4}{3}\pi}$ , or M1 for a correct equation.

(c)

A solid cylinder has radius $x \mathrm{~cm}$ and height $\frac{7 x}{2} \mathrm{~cm}$ .

The surface area of a sphere with radius $R \mathrm{~cm}$ is equal to the total surface area of the cylinder.
Find an expression for R in terms of x.
[The surface area, A, of a sphere with radius r is $A=4 \pi r^{2}$ .]

$\frac{3x}{2}$ or 1.5x or $1\frac{1}{2}x$
B2 for $4R^2=9x^2$ , or M1 for a correct surface-area equation.

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Covered syllabus points

E5.4 Surface area and volume question 1

Question (a)

Question (b)

Question (c)