Corrigé

9.2 Mensuration

Pack A — Réponses

#	Question	Réponse
1	A cuboid has length 8 cm, width 5 cm, height 4 cm. Find its volume.	160 cm³
2	A cube has side 4 cm. Find its total surface area.	96 cm²
3	A cylinder has radius 3 cm and height 10 cm. Find the volume in terms of π.	$90\pi$ cm³
4	A triangle base 6 cm, perpendicular height 8 cm. Find the area.	24 cm²
5	A trapezium has parallel sides $a$ and $b$, height $h$. Find $A$ when $a = 4$, $b = 10$, $h = 5$.	35 cm²
6	A square pyramid has base side 6 cm and height 9 cm. Find the volume.	108 cm³
7	A cone has radius 6 cm, height 12 cm. Find the volume in terms of π.	$144\pi$ cm³
8	A sphere has radius 3 cm. Find the volume in terms of π.	$36\pi$ cm³
9	A cuboid with dimensions 6 × 4 × 3 cm. Find the surface area.	108 cm²
10	A cylinder has radius 4 cm and height 10 cm. Find its total surface area in terms of π (including top and bottom).	$112\pi$ cm²
11	A triangular prism has triangular cross-section with legs 5 and 12 (right-angled), and length 20 cm. Find the volume.	600 cm³
12	A square pyramid has base side 8 cm and slant height 10 cm. Find the surface area.	224 cm²
13	A sphere has radius 6 cm. Find its surface area in terms of π.	$144\pi$ cm²
14	Make $h$ the subject of $V = \pi r^2 h$.	$h = V/(\pi r^2)$
15	A cone has volume $48\pi$ cm³ and radius 4 cm. Find its height.	9 cm
16	Find the volume of a cuboid pyramid (rectangular base) with base 6 × 4 and height 12.	96 cm³
17	A cylinder has volume $200\pi$ cm³ and height 10 cm. Find its radius.	$\sqrt{20} = 2\sqrt{5}$ cm
18	A hemisphere has radius 5 cm. Find (a) volume, (b) curved surface area.	(a) $(250/3)\pi$ (b) $50\pi$
19	A rectangular tank holds 1200 L of water. Length 2 m, width 1.5 m. Find the depth.	0.4 m
20	Convert: a tank holds 250 L. Find its volume in (a) m³ and (b) cm³.	(a) 0.25 m³ (b) 250 000 cm³
21	A triangular prism has a right-angled triangular cross-section with legs 5 cm and 12 cm, and prism length 20 cm. Find its volume.	600 cm³
22	A cuboid has square base side $x$ cm and height 5 cm, SA 192 cm². Find $x$.	$x = 6$
23	Rearrange $V = (1/3)\pi r^2 h$, then find $h$ when $V = 100$ cm³, $r = 5$.	$h = 3V/(\pi r^2) = 12/\pi$ cm
24	A solid metal sphere of radius 6 cm is melted and recast as a cylinder of radius 4 cm. Find the height.	18 cm
25	A cylinder open at the top has radius 3 cm and height 10 cm. Find the surface area (including base, no top).	$69\pi$ cm²
26	A cone has radius 6 cm and slant height 10 cm. Find the (a) curved surface area, (b) total SA.	(a) $60\pi$ cm² (b) $96\pi$ cm²
27	A cone has volume $48\pi$ cm³ and slant height 5 cm. Find the radius and height. (Hint: use Pythagoras for slant.)	$r = 3, h = 4$
28	A composite shape: cylinder topped by a hemisphere. Cylinder r = 4, h = 10. Find total volume in terms of π.	$160\pi + (128/3)\pi = (608/3)\pi$
29	A cuboid has volume 360 cm³. Two sides are 6 and 5. Find the third.	12 cm
30	A swimming pool is a cuboid 25 m by 10 m by 2 m. Find (a) volume in m³, (b) capacity in litres.	(a) 500 m³ (b) 500 000 L
31	A cone has radius 6 cm and height 12 cm. The top is sliced off parallel to the base, producing a smaller cone of radius 3 cm. Find the volume of the remaining frustum (in terms of π).	$126\pi$ cm³
32	A spherical balloon has volume $36\pi$ cm³. (a) Find the radius. (b) The balloon's radius doubles. By what factor does the volume increase?	(a) 3 cm (b) × 8
33	Make $r$ the subject of $V = (4/3)\pi r^3$.	$r = \sqrt[3]{3V/(4\pi)}$
34	A skyscraper modelled as a cuboid is 154 m tall, 87 m wide, 30 m deep. (a) Find the glass area (4 vertical sides). (b) Find the space diagonal.	(a) 36 036 m² (b) ≈ 179.4 m
35	A cylinder has radius $r$ and height $h$ where $h = 2r$. Volume = $54\pi$ cm³. Find $r$.	$r = 3$
36	A solid is formed by adding a cone to a cylinder. The cone and cylinder have the same radius 4 cm. The cylinder is 10 cm tall and the cone is 6 cm tall. Find the total volume.	$160\pi + 32\pi = 192\pi$
37	A hemispherical bowl has volume $\dfrac{2}{3}\pi r^3 = 250$ ml. Find $r$ (give 1 d.p.).	≈ 4.9 cm
38	Two cylinders are similar with linear scale factor 3. The smaller has volume 16 cm³. Find the larger volume.	432 cm³
39	A pyramid has rectangular base 6 × 8 m and apex height 5 m above the centre of the base. Find the slant height to the midpoint of a long side.	√(25 + 9) = √34 ≈ 5.83 m (to midpoint of 8 m side)
40	A cylinder of radius $r$ and height $h$ has surface area $A = 2\pi r^2 + 2\pi r h$. Make $h$ the subject.	$h = (A - 2\pi r^2)/(2\pi r)$

Pack B — Réponses

#	Question	Réponse
1	A cuboid has length 10 cm, width 6 cm, height 3 cm. Find its volume.	180 cm³
2	A cube has side 7 cm. Find its total surface area.	294 cm²
3	A cylinder has radius 5 cm and height 8 cm. Find the volume in terms of π.	$200\pi$ cm³
4	A triangle base 6 cm, perpendicular height 8 cm. Find the area.	60 cm²
5	A trapezium has parallel sides $a$ and $b$, height $h$. Find $A$ when $a = 4$, $b = 10$, $h = 5$.	40 cm²
6	A square pyramid has base side 6 cm and height 9 cm. Find the volume.	64 cm³
7	A cone has radius 6 cm, height 12 cm. Find the volume in terms of π.	$192\pi$ cm³
8	A sphere has radius 6 cm. Find the volume in terms of π.	$288\pi$ cm³
9	A cuboid with dimensions 6 × 4 × 3 cm. Find the surface area.	132 cm²
10	A cylinder has radius 4 cm and height 10 cm. Find its total surface area in terms of π (including top and bottom).	$60\pi$ cm²
11	A triangular prism has triangular cross-section with legs 5 and 12 (right-angled), and length 20 cm. Find the volume.	360 cm³
12	A square pyramid has base side 8 cm and slant height 10 cm. Find the surface area.	144 cm²
13	A sphere has radius 6 cm. Find its surface area in terms of π.	$100\pi$ cm²
14	Make $h$ the subject of $V = \pi r^2 h$.	Same.
15	A cone has volume $48\pi$ cm³ and radius 4 cm. Find its height.	12 cm
16	Find the volume of a cuboid pyramid (rectangular base) with base 6 × 4 and height 12.	120 cm³
17	A cylinder has volume $200\pi$ cm³ and height 10 cm. Find its radius.	2 cm
18	A hemisphere has radius 5 cm. Find (a) volume, (b) curved surface area.	(a) $144\pi$ (b) $72\pi$
19	A rectangular tank holds 1200 L of water. Length 2 m, width 1.5 m. Find the depth.	0.5 m
20	Convert: a tank holds 250 L. Find its volume in (a) m³ and (b) cm³.	(a) 5 m³ (b) 5 000 000 cm³
21	A triangular prism has a right-angled triangular cross-section with legs 6 cm and 8 cm, and prism length 15 cm. Find its volume.	360 cm³
22	A cuboid has square base side $x$ cm and height 3 cm, SA 80 cm². Find $x$.	$x = 4$
23	Rearrange $V = (1/3)\pi r^2 h$, then find $h$ when $V = 100$ cm³, $r = 5$.	$h = 18/\pi$ cm
24	A solid metal sphere of radius 6 cm is melted and recast as a cylinder of radius 4 cm. Find the height.	27 cm
25	A cylinder open at the top has radius 3 cm and height 10 cm. Find the surface area (including base, no top).	$80\pi$ cm²
26	A cone has radius 6 cm and slant height 10 cm. Find the (a) curved surface area, (b) total SA.	(a) $65\pi$ cm² (b) $90\pi$ cm²
27	A cone has volume $48\pi$ cm³ and slant height 5 cm. Find the radius and height. (Hint: use Pythagoras for slant.)	Same.
28	A composite shape: cylinder topped by a hemisphere. Cylinder r = 4, h = 10. Find total volume in terms of π.	$54\pi + 18\pi = 72\pi$
29	A cuboid has volume 360 cm³. Two sides are 6 and 5. Find the third.	10 cm
30	A swimming pool is a cuboid 25 m by 10 m by 2 m. Find (a) volume in m³, (b) capacity in litres.	(a) 675 m³ (b) 675 000 L
31	A cone has radius 6 cm and height 12 cm. The top is sliced off parallel to the base, producing a smaller cone of radius 3 cm. Find the volume of the remaining frustum (in terms of π).	$224\pi$ cm³
32	A spherical balloon has volume $36\pi$ cm³. (a) Find the radius. (b) The balloon's radius doubles. By what factor does the volume increase?	Same.
33	Make $r$ the subject of $V = (4/3)\pi r^3$.	Same.
34	A skyscraper modelled as a cuboid is 154 m tall, 87 m wide, 30 m deep. (a) Find the glass area (4 vertical sides). (b) Find the space diagonal.	(a) 31 200 m² (b) ≈ 152.6 m
35	A cylinder has radius $r$ and height $h$ where $h = 2r$. Volume = $54\pi$ cm³. Find $r$.	$r = 4$
36	A solid is formed by adding a cone to a cylinder. The cone and cylinder have the same radius 4 cm. The cylinder is 10 cm tall and the cone is 6 cm tall. Find the total volume.	$72\pi + 18\pi = 90\pi$
37	A hemispherical bowl has volume $\dfrac{2}{3}\pi r^3 = 250$ ml. Find $r$ (give 1 d.p.).	≈ 6.2 cm
38	Two cylinders are similar with linear scale factor 3. The smaller has volume 16 cm³. Find the larger volume.	200 cm³
39	A pyramid has rectangular base 6 × 8 m and apex height 5 m above the centre of the base. Find the slant height to the midpoint of a long side.	√(36 + 4) = √40 ≈ 6.32 m
40	A cylinder of radius $r$ and height $h$ has surface area $A = 2\pi r^2 + 2\pi r h$. Make $h$ the subject.	Same.

Problèmes — Solutions détaillées

**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.

Réponse

(a) $96\pi$ (b) $\tfrac{8}{3}$ cm

(a) $V = (1/3)\pi(36)(8) = 96\pi$. (b) $\pi(36) h = 96\pi \Rightarrow h = 96/36 = 8/3$ cm ≈ 2.67 cm.

**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.

Réponse

(a) ≈ 2 592 100 m³ (b) ≈ 219.2 m

(a) $V = (1/3)(230)^2(147) = (1/3)(52900)(147) = 2 592 100$ m³. (b) Half-diagonal of base: $230\sqrt{2}/2 \approx 162.6$ m. Slanted edge = $\sqrt{147^2 + 162.6^2} \approx 219.2$ m.

**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.).

Réponse

(a) ≈ 50 m (b) ≈ 519 000 m³

(a) $4\pi r^2 = 31416 \Rightarrow r^2 = 2500 \Rightarrow r = 50$ m. (b) Internal $r = 49.85$ m. Internal volume = $(4/3)\pi (49.85)^3 \approx 519 000$ m³.

**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.

Réponse

≈ 19 cm (2 s.f.)

Tank vol = 72 000 cm³. Water = 24 000 cm³. Cylinder base $= \pi(400) \approx 1257$. Depth = $24000/1257 \approx 19$ cm.

**Rearranging formulas.** $V = (4/3)\pi r^3$. (a) Make $r$ the subject. (b) Find $r$ for $V = 36\pi$.

Réponse

(a) $r = \sqrt[3]{3V/(4\pi)}$ (b) $r = 3$

(a) Multiply by 3, divide by $4\pi$: $r^3 = 3V/(4\pi)$, $r = \sqrt[3]{3V/(4\pi)}$. (b) $r^3 = 3(36\pi)/(4\pi) = 27 \Rightarrow r = 3$.

**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 $x$.

Réponse

(a) 4 cm (b) $x = 20$ cm

(a) Height equals the cut depth = 4 cm. (b) $V = x^2 \times 4 = 1600 \Rightarrow x = 20$ cm.

**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.

Réponse

(a) 3 cm (b) ≈ 280π cm³ — see working

(a) Similar triangles: at height 8, distance to apex = 4 above the cut. Radius shrinks linearly from 9 (at base) to 0 (at apex). At height 8: radius = $9 \times (12 - 8)/12 = 9 \times 1/3 = 3$. (b) Big cone vol = $(1/3)\pi(81)(12) = 324\pi$. Small (top sliced off) cone vol = $(1/3)\pi(9)(4) = 12\pi$. Frustum = $324\pi - 12\pi = 312\pi$. Hmm — let me recheck: the slice is at height 8 cm above the base. The small cone above the slice has height $12 - 8 = 4$ cm and radius 3. Vol $= (1/3)\pi(9)(4) = 12\pi$. Frustum = $324\pi - 12\pi = 312\pi$ cm³.

**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).

Réponse

(a) $250\pi + (250/3)\pi = (1000/3)\pi$ (b) Cyl side $100\pi$ + base $25\pi$ + hemi $50\pi = 175\pi$

(a) Cyl vol = $\pi(25)(10) = 250\pi$. Hemi vol = $(2/3)\pi(125) = (250/3)\pi$. Total $= (1000/3)\pi$ cm³. (b) Curved cyl side = $2\pi(5)(10) = 100\pi$. Base circle = $25\pi$. Hemi surface = $2\pi(25) = 50\pi$. Total = $175\pi$.

**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.

Réponse

(a) 4 cm (b) $4\sqrt{10}$ cm (c) ≈ 71.6°

(a) G is centre of square base; M is midpoint of QR (a side). GM = half-side = 4. (b) $VM = \sqrt{12^2 + 4^2} = \sqrt{160} = 4\sqrt{10}$ cm. (c) Angle VMG: $\tan = 12/4 = 3$. $\theta = \tan^{-1}(3) \approx 71.6°$.

**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?

Réponse

(a) $(160/3)\pi$ cm³ ≈ 167.6 (b) ≈ 3.35 s (c) No — glass holds $200\pi$

(a) $V = (1/3)\pi(16)(10) = (160/3)\pi \approx 167.6$ cm³. (b) $167.6/50 \approx 3.35$ s. (c) Glass cap $= \pi(25)(8) = 200\pi \approx 628$ cm³. Funnel is much smaller — no overflow.

**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.

Réponse

(a) $1728 + 144\pi \approx 2180.4$ cm³ (b) Cube SA (one face replaced) + hemi curved + circle ≈ see working

(a) Cube $V = 12^3 = 1728$. Hemi $V = (2/3)\pi(216) = 144\pi \approx 452$. Total $\approx 2180$ cm³. (b) Cube has 6 faces of $144$. Top face replaced by hemisphere curved $2\pi(36) = 72\pi$. Plus the circular bottom of the hemisphere (visible if the cube is hollow), but assumed not exposed. SA = 5 faces of cube + curved hemi = $5(144) + 72\pi = 720 + 72\pi \approx 946$ cm².

**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.

Réponse

(a) 8 (b) 400 (c) 36

(a) Volume scale factor = $2^3 = 8$. (b) $50 \times 8 = 400$ cm³. (c) SA scale factor = $2^2 = 4$. Smaller = $144/4 = 36$ cm².