Problem-solving Pack

Mathematics

Year 9 · 9.2 Mensuration

Problem-solving Pack

Name: _________________________________

Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12

Cone and cylinder. A cone has radius 6 cm and height 8 cm. A cylinder has the same radius and is made of the same volume of material.

(a) Find the volume of the cone (in terms of π).
(b) Find the height of the cylinder.

Working space

2Problem 2 of 12

Pyramid at Giza. A square-based pyramid has base side 230 m and vertical height 147 m.

(a) Find the volume.
(b) Find the length of one slanted edge from a base vertex to the apex.

Working space

3Problem 3 of 12

Surface area of a sphere. A glass sphere has external SA ≈ 31 416 m².

(a) Find the external radius.
(b) If the glass is 15 cm thick, find the internal volume (3 s.f.).

Working space

4Problem 4 of 12

Fish tank. A tank 60 × 40 × 30 cm is 33% full. Water is poured into a cylindrical container with radius 20 cm. Find the depth of water in the cylinder.

Working space

5Problem 5 of 12

Rearranging formulas.

V = (4/3)\pi r^3

.

(a) Make

r

the subject.
(b) Find

r

for

V = 36\pi

Working space

6Problem 6 of 12

Pizza box from a square sheet. Six 4 cm × 4 cm squares are cut from corners of a square sheet, which is then folded into a square prism (open-top). The base is

x

cm ×

x

cm.

(a) Height of box = ?
(b) Volume = 1600 cm³. Find

x

Working space

7Problem 7 of 12

Volume of a frustum. A cone has radius 9 and height 12 cm. The top is sliced parallel to the base at a height of 8 cm, leaving a frustum.

(a) Find the radius of the new top.
(b) Find the volume of the frustum.

Working space

8Problem 8 of 12

Cylinder + hemisphere. A solid is a cylinder with a hemisphere on top, both of radius 5 cm. Cylinder height = 10 cm.

(a) Find the total volume.
(b) Find the total surface area (don't count the joined circle).

Working space

9Problem 9 of 12

Trigonometry on a square pyramid. PQRS is the square base of a pyramid with apex V. Square sides 8 cm; vertical height VG = 12 cm; M is the midpoint of QR.

(a) Write down GM.
(b) Find VM.
(c) Find the angle between face VQR and the base.

Working space

10Problem 10 of 12

Capacity puzzle. A cone-shaped funnel has top radius 4 cm and height 10 cm. Water pours in from a tap at 50 cm³/s.

(a) Find the funnel's capacity.
(b) Find the time to fill the funnel.
(c) The funnel pours into a cylindrical glass (r = 5, h = 8). Will the glass overflow when the funnel fully empties into it?

Working space

11Problem 11 of 12

Composite shape volume. A solid is composed of a hemisphere of radius 6 cm on top of a cube of side 12 cm.

(a) Find the total volume.
(b) Find the total exposed surface area.

Working space

12Problem 12 of 12

Volume scaling. Two similar pyramids have a linear scale factor of 2.

(a) Find the volume scale factor.
(b) The smaller has volume 50 cm³. Find the larger volume.
(c) The larger has surface area 144 cm². Find the smaller surface area.

Working space