CAIE IGCSE Mathematics Extended Mensuration Question Bank

Practise measurement units, perimeter, area, circles, sectors, surface area and volume across two- and three-dimensional mensuration problems.

Covered syllabus points

E5.1 Units of measure E5.2 Area and perimeter E5.3 Circles, arcs and sectors E5.4 Surface area and volume E5.5 Compound shapes and parts of shapes

[Maximum number: 9]

(a)

NOT TO SCALE

Calculate the area of the trapezium.

\text { ........................................ } .1^{2}

[ 2 ]

88
M1 for $\frac{1}{2}(9+13)\times8$ oe

(b)

NOT TO SCALE

The diagram shows a sector of a circle with radius 8 cm and angle $75^{\circ}$ .
Find the perimeter of the sector.
cm

[ 3 ]

26.5 or 26.47...
B2 for 10.5 or 10.47... or $\frac{10\pi}{3}$
OR M2 for $8+8+\frac{75}{360}\times2\pi\times8$
or M1 for $\frac{75}{360}\times2\pi\times8$

(c)

Figure for Question (c) — CAIE IGCSE Mathematics Extended

NOT TO SCALE

The diagram shows a shape ABQP made from three straight lines and an arc of a sector of a circle. The sector has centre O and angle $90^{\circ}$ .
POQ is a straight line and $A P=P O=O Q=Q B=5 \mathrm{~cm}$ .
Find the area of ABQP.
Give your answer in the form $a+k \pi$ .
$\mathrm{cm}^{2}$

[ 4 ]

$25+\frac{25\pi}{2}$
M2 for $\frac{90}{360}\times\pi\times(\sqrt{5^2+5^2})^2$
or M1 for $\text{radius}^2=5^2+5^2$
M1 for triangle area $[2\times]\frac{1}{2}\times5\times5$ oe

[Maximum number: 4]

NOT TO SCALE

A, B and C are points on the circle, centre O.
DE is a tangent to the circle at C.
AC=10 cm, AB=9.5 cm and BC=7.7 cm.

Calculate the area of triangle ABC as a percentage of the area of the circle.

38.8 or 38.9 or 38.78 to 38.85
M3 for $\frac{0.5\times9.5\times7.7\times\sin70.2}{\pi(\text{their (c)})^2}\times100$ oe
or lower method marks as in markscheme

[Maximum number: 2]

(a)

Figure for Question (a) — CAIE IGCSE Mathematics Extended

NOT TO SCALE

Triangle ABC is mathematically similar to triangle PQR.
The area of triangle ABC is $16 \mathrm{~cm}^{2}$ .

[ 2 ]

(i)

Calculate the area of triangle PQR.
$\mathrm{cm}^{2}$

[ 2 ]

36
M1 for scale-area factor $\left(\frac{8}{12}\right)^2$ or $\left(\frac{12}{8}\right)^2$ .

(b)

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.

(c)

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.

(d)

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.

[Maximum number: 9]

(a)

ABCDEFGH is a regular octagon with sides of length 6 cm. The diagram shows part of the octagon.
O is the centre of the octagon and M is the midpoint of AB.

NOT TO SCALE

[ 3 ]

(i)

Find the area of the circle that passes through the vertices of the octagon.
$\mathrm{cm}^{2}$

[ 3 ]

193 or 193.0 to 193.1
M2 for $\pi\left(\frac3{\cos67.5^\circ}\right)^2$ oe
or M1 for $\frac3r=\cos67.5^\circ$ or $\frac{\sin45^\circ}{6}=\frac{\sin67.5^\circ}{r}$

(b)

Figure for Question (b) — CAIE IGCSE Mathematics Extended

The diagram shows a horizontal container for water with a uniform cross-section. The cross-section is a semicircle.
The radius of the semicircle is 0.45 m and the length of the container is 4 m .

[ 6 ]

(i)

NOT TO SCALE

The greatest depth of the water in the container is $0.3\,\mathrm{m}$. The diagram shows the cross-section.
Calculate the number of litres of water in the container.
Give your answer correct to the nearest integer.

[ 6 ]

Figure for Answer to Question (i) — CAIE IGCSE Mathematics Extended

742 or 743
M5 for a method leading to the volume of water, e.g.
$4\left\{2\times\frac{\operatorname{invcos}(0.15/0.45)}{360}\times\pi\times0.45^2-\frac12\times0.45^2\times\sin\left(2\operatorname{invcos}(0.15/0.45)\right)\right\}$
OR M2 for a correct sector area expression and M2 for a correct triangle area expression
If 0 scored, SC1 for $\operatorname{invcos}(0.15/0.45)$ , $\operatorname{invsin}(0.15/0.45)$ , or $\sqrt{0.45^2-0.15^2}$ soi

All question bank results loaded

Covered syllabus points

5. Mensuration question 1

Question (a)

Question (b)

Question (c)

5. Mensuration question 2

5. Mensuration question 3

Question (a)

Question (i)

Question (b)

Question (c)

Question (d)

5. Mensuration question 4

Question (a)

Question (i)

Question (b)

Question (i)