1.Find the Unknown in Each of the Following Figures. (Correct Your Answers to 3 Significant

Chapter 8Mensuration 8.1

Chapter 8Mensuration

Warm-up Exercise

1.Find the unknown in each of the following figures. (Correct your answers to 3 significant figures if necessary.)

(a)(b)(c)

2.Find the volume of each of the following solids. (Express your answers in terms of  if necessary.)

(a)(b)(c)

3.Find the total surface area of each of the following solids. (Correct your answers to 3significant figures if necessary.)

(a)(b)(c)

4.If x:y:z1:4:6,

(a)(i)expressy in terms of x.

(ii)express z in terms of x.

(b)find x:xy:xyz.

5.(a)Given that y3x, if x increases by 5%, find the percentage increasein y.

(b)Given that y5x3, if x decreases by20%, find the percentage decrease in y.

6.In each of the following figures, ABCADE. Find the unknowns.

(a)(b)(c)

Build-up Exercise

[This part provides two extra sets of questions for each exercise in the textbook, namely Elementary Setand Advanced Set. You may choose to complete any ONE set according to your need.]

Exercise 8A

[In this exercise, unless otherwise stated, correct your answers to 3 significant figures if necessary.]

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Level 1

1.Complete the following table.

Shape of the base of a pyramid / Height / Base area / Volume
(a) / / 4cm
(b) / / 5cm
(c) / / 6cm
(d) / / 8cm
(e) / / 12cm
2.The base of a pyramid is an isosceles right-angled triangle where the lengths of the two equal sides are 8cm. The height VE of the pyramid is 15cm. Find the volume of the pyramid. /
3.In the figure, VABC is a pyramid. ABC is an isosceles right-angled triangle.IfABAC40cmand VBVC50cm,find the volume of pyramid VABC. /

4.The height and volume of a pyramid are 12cm and 120cm3respectively. Its base is a rectangle with dimensions 6cmxcm. Find x.

5.In the figure, VABCD is a right pyramid. Its base ABCD is a square with sides of 6cm each. E is a point on BC such that VEBC and VE10cm. Find the total surface area of the pyramid. /

Level 2

6.In the figure, VABCD is a right pyramid. The base ABCD is a square with sides of 5cm each. The slant edge is 8cm long.
(a)Find the height VO of the pyramid.
(b)Find the volume of the pyramid. /
7.The figure shows the net of a right pyramid where the base is a square with sides of 16cmeach.
(a)Find the total surface area of the pyramid.
(b)Find the height of the pyramid.
(c)Find the volume of the pyramid. /
8.The figure shows a frustum with right-angled triangular bases where AC15cm, AB12cm, PQ12cm and AP10cm.
(a)By using similar triangles VPQ and VAC, find VA.
(b)By using similar triangles VPR and VAB, find PR.
(c)Hence, find the volume of the frustum ABCQPR. /

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Level 1

1.The figure shows a right pyramid with a regular hexagonal base. The base area and height of the pyramid are 50cm2 and 6cm respectively. Find the volume of the pyramid. /

2.It is given that the base of a pyramid is a triangle with base acm and height bcm. If the height of the pyramid is hcm, express the volume of the pyramid in terms of a, b and h.

3.In the figure, ABCD is a trapezium where AB16cm, AD10cm and CD20cm.
(a)Find the area of trapezium ABCD.
(b)If trapezium ABCD is a base of a pyramid with a height of 20cm, find the volume of the pyramid. /

4.The height and volume of a pyramid are 12cm and 100cm3respectively. If the base of the pyramid is a square, find the length of each side of the square base.

Level 2

5.The base of a right pyramid is a square with an area of 81cm2. The height is 15cm. Find the length of the slant edge of the pyramid.

6.In the figure, VABC is a pyramid. ABC is an isosceles right-angled triangular base where ABAC30cm. The height VA of the pyramid is 20cm.
(a)Find the area of VBC.
(b)Find the total surface area of the pyramid. /
7.In the figure, VABCD is a right pyramid where the base is a rectangle with dimensions 24cm10cm. The slant edge is 30cm long.
(a)Find the height VE of the pyramid.
(b)Find the volume of the pyramid.
(c)Find the total surface area of the pyramid. /
8.In the figure, ABCDEFGH is a cuboid with the height of 50cm.Its base is a square with dimensions 20cm20cm.VEFGH is a right pyramid with the same height as the cuboid.
(a)Find the total surface area of pyramid VEFGH.
(b)If pyramid VEFGH is removed from the cuboid, find the total surface area of the remaining solid. /
9.If a solid metallic square-based pyramid is melted and recast to form another square-based pyramid which is 21% higher than the original pyramid, find the percentage decrease in the length of each side of the square base. /
10.In the figure, VABCD is a square-based right pyramid. After removing right pyramid VPQRS from pyramid VABCD, a frustum with square bases is formed. It is given that AB20cm, PQ12cm and PA8cm.
(a)By considering similar triangles VPQ and VAB, find VA and the height of the pyramid VABCD.
(b)Hence, find the volume of frustum.
(c)Find the height of VAB from point V.
(d)Hence, findthe total surface area of frustum PQRSDABC. /

Exercise 8B

[In this exercise, unless otherwise stated, correct your answers to 3significant figures if necessary.]

Elementary Set

Level 1

1.Find the volume of each of the following right circular cones. (Express your answers in terms of .)

(a)(b)(c)

2.Find the curved surface area of each of the following right circular cones. (Express your answers in terms of .)

(a)(b)(c)

3.Both the diameter and slant height of a right circular cone are 24cm.

(a)Find the total surface area of the cone in terms of .

(b)Find the volume of the cone.

4.The figure shows an inverted right conical paper cup. The capacity of the paper cup is 180cm3and the base radius is 4cm.
(a)Find the height of the paper cup.
(b)If the cup is fully filled with water, find the area of the wet surface. /
5.The figure shows an inverted right conical popcorn cup formed by rolling up a paper sector. It is given that the slant height of the popcorn cup is 20cm, and the perimeter of the base is 50cm.
(a)Find the area of the paper sector.
(b)If the cost of paper for making the popcorn cup is $6/m2, find the cost of paper for making 200 popcorn cups. /

Level 2

6.300 pieces of identical conical chocolate are made from 1000cm3 of chocolate. If the height of each piece of chocolate is 1cm, find its base radius.

7.If a right circular cone is formed by rolling up the sector as shown,
(a)find the base radius of the cone.
(b)find the height of the cone.
(c)find the volume of the cone. /
8.The spinning top shown in the figure is formed by three parts. I and II are right cylinders. III is an inverted right circular cone.
(a)Find the volume of the spinning top in terms of .
(b)Find the total surface area of the spinning top. /
9.A right circular frustum is formed by rotating trapezium ABCD 360 aboutthe axis AD. It is given that AB6cm, AD9cm and DC15cm.
(a)Find the volume of the frustum in terms of .
(b)Find the total surface area of the frustum. /
10.The figure shows an inverted right conical funnel with the base radius of 5cm and height of 16cm. Initially, the funnel is fully filled with water. After a while, the water level drops to 8cm.
(a)Find the radius of the water surface.
(b)What percentage of water is dripped from the funnel? /

Advanced Set

Level 1

1.Find the volume and total surface area of each of the following right circular cones. (Express your answers in terms of .)

(a)(b)(c)

2.The slant height of a right circular cone is 18cm and the height is half of the slant height.

(a)Find the volume of the cone in terms of .

(b)Find the total surface area of the cone.

3.The figure shows a right circular conical hat formed by rolling up a paper sector. It is given that the slant height of the hat is 25cm and the perimeter of the base is 18cm.
(a)Find the area of the paper sector in terms of .
(b)If the cost of paper for making the hat is $10/m2, find the cost ofpaper for making 50conical hats. /

Level 2

4.(a)A metallic right cylinder with both base radius and height of 10cm is melted.of the metal is recast to form a right circular cone with the base same as the original cylinder. Find the height of the cone.

(b)The remaining metal is recast to form another right circular conewith the base same as the original cylinder. Find the total surface area of this cone.

5.The figure shows an ice-cream cone where the volume of the ice-cream is 400cm3. The height of the cone is 12cm and it is filled with ice-cream. The ratio of the volume of ice-cream outside the cone to that inside the cone is 35,find the radius of the ice-cream cone. /
6.The figure shows a chocolate in the shape of a right circular frustum. The upper and lower base diameters are 2cm and 3cm respectively.
(a)Find the volume of the chocolate.
(b)It is given that every cm3 of chocolate weighs 3g. How many chocolates as shown in the figure can be produced from 1kg of chocolate? /
7.The figure shows a paper sector with an area of 120cm2. If a right circular cone is formed by rolling up the paper sector,
(a)find the base radius of the cone.
(b)find the height of the cone.
(c)find the volume of the cone. /
8.The figure shows an inverted right conical cup containing 8cm3of water. The diameter of the water surface is 4cm and the water surface is 2cm below the rim of the cup.
(a)Find the depth of water.
(b)Find the area of the wet surface.
(c)Find the capacity of the cup. /

9.Figure A shows a right circular paper cone where the base radius and height are 7cm and 24cm respectively. When the cone shown in figure A is cut along a slant height, a sector shown in figure B is formed. Two such sectors are joined to form the sector in figure C, and then rolled up to form a right circular cone.

(a)Find the base radius of the new cone.

(b)Find the height of the new cone.

(c)Find the volume of the new cone. Is the volume of the new cone twice that of the cone in figure A?

10.The figure shows a rocket model made up of three parts. Solid I is a right circular cone. Solid II is a right cylinder. Solid III is a right circular frustum.
(a)Find the volume of solid III in terms of .
(b)Find the volume of the rocket model in terms of .
(c)Find the total surface area of the rocket model. /

Exercise 8C

[In this exercise, unless otherwise stated, express your answers in terms of  if necessary.]

Elementary Set

Level 1

1.Find the value of x in each of the following. (Correct your answers to 3significant figures if necessary.)

(a)x327(b)x310(c)x3

2.The radius of a sphere is 6cm.
(a)Find the volume of the sphere.
(b)Find the surface area of the sphere. /
3.The diameter of a sphere is 8cm.
(a)Find the volume of the sphere.
(b)Find the surface area of the sphere. /
4.If the base area of a hemisphere is 54cm2, find the curved surface area of the hemisphere. /

5.If the volume of a sphere is 10cm3, find the radius of the sphere. (Correct your answer to 3 significant figures.)

6.If the volume of a sphere is, find the surface area of the sphere.

7.If the surface area of a sphere is 64cm2, find the volume of the sphere.

Level 2

8.In the figure, Ois the centre of the circle, the circumference is 36cm. A sphere is formed by rotating the circle 360 about diameter AOB.
(a)Find the surface area of the sphere.
(b)Find the volume of the sphere. /

9.A metal hemisphere with the radius of 4cm is melted and recast to form a metal sphere.

(a)Determine whether the total surface area of the solid is increased or decreased.

(b)Find the percentage increase / percentage decrease in the total surface area of the solids. (Correct your answer to 3 significant figures.)

10.The figure shows a right cylindrical container with water. After putting a number of metal balls with diameters of 1cm each into the container, the water level rises 2cm. Assume that all the metal balls are fully immersed in water and water does not overflow, find the number of metal balls put in the water. /
11.A pill with the length of 15mm is shown in the figure. If both ends of the pill are hemispheres,
(a)find the volume of the pill.
(b)find the total surface area of the pill. /

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Level 1

1.Find the value of y in each of the following. (Correct your answers to 3significant figures if necessary.)

(a)y364(b)y325(c)y327

2.Find the volume and total surface area of each of the following solids.

(a)(b)(c)

3.If the volume of a sphere is 100cm3, find the diameter of the sphere. (Correct your answer to 3 significant figures.)

4.A hemispherical pudding with the volume of 144cm3 is shown in the figure. Find its total surface area. /

5.If the surface area of a sphere is 3600cm2, find the volume of the sphere.

Level 2

6.If the outer diameter of a hollow metal sphere is 12mm and the thickness is 2mm, find the volume of metal required to form the metal sphere.

7.The figure shows a sculpture formed by two hemispheres. The upper part is a hemisphere with the radius of 0.4m and the lower part is a hemisphere with the radius of 1m.
(a)Find the volume of the sculpture.
(b)Find the total surface area of the sculpture. /

8.A few years ago, the standard diameter of a table tennis ball for competition changed from 38mm to 40mm. Find the percentage increase in the surface area of a table tennis ball for competition. (Correct your answer to 3 significant figures.)

9.An inverted right pyramid is removed from a solid hemisphere, where the square base of the pyramid has been inscribed in the base of the hemisphere. Given that the slant edge of the pyramid is 5cm long, find the volume of the remaining solid. (Correct your answer to 3 significant figures.) /
10.In the figure, a metal ball with the radius of 5cm is inside a container in theshape of right prism. The base of the container is a rectangle with dimensions 15cm12cm.Water is poured into the container until the metal ball is just covered by water.
(a)Find the volume of water.
(b)Now, 10 more metal balls with diameters of 2.4cm each are put into the container. Assume that the metal balls are fully immersed in water and water does not overflow, how much does the water level rise?
(Correct your answers to 3 significant figures.) /

11.Figure A shows a hemisphere with the radius of rcm. Figure B shows a solid which is formed by removing an inverted right circular cone from a right circular cylinder.The base radii and heights of the cone and cylinder are allrcm.

(a)Show that the volumes of solids in figures A and B are equal.

(b)Figures C and D show the cross-sections of figures A and B respectively, and both of them are zcm from the bases. Show that the areas of the two cross-sections are equal.

Exercise 8D

[In this exercise, unless otherwise stated, correct your answers to 3 significant figures if necessary.]

Elementary Set

Level 1

1.Which of the following formulae is the one for the area of the regular pentagon as shown?
I.
II.
III. /
2.Thefigure shows a right frustum with square bases. Which of the following formulae is the one for the total surface area of the frustum? Which one is the formula for its volume?
I.
II. /
3.A and B are two uniform cross-sections of two similar right prisms.
(a)Find the ratio of the total surface area of the small prism to that of the large prism.
(b)Find the ratio of the volume of the small prism to that of the large prism. /

4.The figure shows two similar solids A and B.

(a)Find the ratio of the total surface area of solid A to that of solid B.

(b)Find the ratio of the volume of solidA to that of solid B.

5.The ratio of the length of a model car to that of the real car is1:24. If the area of the windscreen of the real car is 1m2, find the area of the windscreen of the model car in cm2.

6.In the figure, A and B are two similar solids. If the area of the cross-section of solid A is 84cm2, find the area of the cross-section of solid B. (Express your answer in terms of .) /
7.In the figure, A and B are two similar solids. If the volume of solid B is 6cm3, find the volume of solid A. /

Level 2

8.According to the following ratios of the volumes V1:V2 of similar solids, find the ratios of their corresponding lengthsl1:l2 and the ratios of their total surface areas A1:A2.

(a)V1:V2125:512(b)V1:V264:27

9.According to the following ratios of the total surface areas A1:A2 of similar solids, find the ratios of their corresponding lengths l1:l2 and the ratios of their volumes V1:V2.

(a)A1:A24:25(b)A1:A2121:169

10.In the figure, the top of the solid is a square with sides of 6cm each. The volume of the solid is 792cm3. If a similar solid is produced such that the area of the top is 2.25 times of the given one, find the volume of the new solid. /

11.If the radius of a spherical balloon increases by 15%, find the percentage increase in the volume of the balloon.

12.The figure shows an inverted right conical paper cup with water. The depth of water is 5cm. After drinking half of the water, what is the depth of water? /

Advanced Set

Level 1

1.In the following formulae, which one is the formula for the area of the ellipse as shown?
I.
II.
III. /
2.Which of the following formulae is the one for the total surface area of the regular icosahedron as shown? Which one is the formula for its volume?
I.
II. /
3.A and B are two uniform cross-sections of two similar prisms.
(a)Find the ratio of the total surface area of the large prism to that of the small one.
(b)Find the ratio of the volume of the large prism to that of the small one. /
4.A and B are two similar bowling pins.
(a)Find the ratio of the total surface area of the small bowling pin to that of the large one.
(b)Find the ratio of the volume of the small bowling pin to that of the large one. /

5.The volume ofa figure of Bruce Lee with heightof his real heightis 600cm3.If a similar bronze statue is produced with its height equals 1.5 times the real height of Bruce Lee, what is its volume?