CAIE IGCSE Mathematics Extended Advanced Surface Area and Volume

Practise solving linked solid mensuration problems with slant heights, equal volumes or areas, recast material and capacity conversions.

Exam points

find missing slant or perpendicular dimensions before substituting into surface-area or volume formulas
set total volumes or surface areas equal when a solid is melted, recast or compared algebraically
calculate the number of complete solids by dividing available volume and rounding down appropriately

(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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E5.4.1 Carry out calculations and solve problems involving the surface area question 1

Question (a)

Question (b)

Question (c)