Volume And Surface Area

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課程筆記

Volume of Cubes and Cuboids

Volume of a cuboid: V = lwh (length × width × height).
A cube is a special cuboid with all sides equal: V = s³.
Terms like 'depth' or 'breadth' may be used instead of height or width.

Cuboid

Volume of Prisms

Volume of a prism: V = A × l where A is the cross-sectional area and l is the length.
The cross-section is constant throughout the prism.
If the volume and length are known, the cross-sectional area can be found by rearranging: A = V / l.

Prism

Volume of Cylinders

Volume of a cylinder: V = πr²h (given in exam).
A cylinder is like a prism with a circular cross-section.

Cylinder

Volume of Pyramids and Cones

Volume of a pyramid: V = (1/3) × base area × perpendicular height.
Volume of a cone: V = (1/3)πr²h (given in exam).
The height must be perpendicular to the base.

Pyramid

Volume of Spheres

Volume of a sphere: V = (4/3)πr³ (given in exam).
A hemisphere is half a sphere: V = (2/3)πr³.

Sphere

Problem Solving with Volumes

For compound shapes, find volumes of individual parts and add or subtract.
For frustums (truncated cones/pyramids), subtract the smaller volume from the larger.
Always check units and convert if necessary (e.g., cm to m, litres).
Real-world problems often combine volume with cost or capacity.

Frustum of a cone

Surface Area of Cubes, Cuboids, Prisms, and Pyramids

Surface area is the sum of the areas of all faces.
For flat-faced solids, calculate each face area separately and add.
Drawing a net can help visualise all faces.

Net of a pyramid

Surface Area of Cylinders

Curved surface area: A = 2πrh (given in exam).
Total surface area: A = 2πrh + 2πr² (not given).
The net consists of two circles and a rectangle.

Net of a cylinder

Surface Area of Cones

Curved surface area: A = πrl where l is slant height (given in exam).
Total surface area: A = πrl + πr² (not given).
Slant height l can be found using Pythagoras: l = √(r² + h²).

Net of a cone

Surface Area of Spheres and Hemispheres

Surface area of a sphere: A = 4πr² (given in exam).
Hemisphere curved surface area: 2πr²; total (including base): 3πr².

Surface area of a hemisphere

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練習題

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1.What is the formula for the volume of a cuboid with length l, width w, and height h?
Easy
- AV = lwh
- BV = l + w + h
- CV = 2lw + 2wh + 2lh
- DV = l² + w² + h²
2.The volume of a cube is 1000 cm³. What is the side length of the cube?
Easy
- A10 cm
- B100 cm
- C33.3 cm
- D31.6 cm
3.A solid cylinder has radius 3 cm and height 4.5 cm. Calculate its total surface area in terms of π.
Medium
- A45π cm²
- B27π cm²
- C54π cm²
- D36π cm²
4.The volume of a cuboid is 180 cm³ and its base is a square of side 6 cm. What is its height?
Easy
- A5 cm
- B30 cm
- C3 cm
- D6 cm
5.A hemisphere has radius 6 cm. Calculate its volume. Give your answer in terms of π.
Medium
- A144π cm³
- B288π cm³
- C72π cm³
- D216π cm³
6.A solid metal cone has radius 10 cm and height 36 cm. Calculate its volume. Give your answer in terms of π.
Medium
- A1200π cm³
- B3600π cm³
- C120π cm³
- D360π cm³
7.A solid cylinder and a solid sphere have the same radius r. The cylinder has height 8r. What fraction of the cylinder's volume is the sphere's volume?
Medium
- A1/6
- B1/3
- C1/2
- D2/3
8.The surface area of a sphere is 81π cm². Find the curved surface area of a cylinder with radius 3r and height 8r, where r is the sphere's radius.
Hard
- A972π cm²
- B216π cm²
- C108π cm²
- D324π cm²

更多題目

9.A solid cylinder has radius x cm and height 7x/2 cm. The total surface area of the cylinder equals the surface area of a sphere of radius R cm. Find R in terms of x.
Hard
- AR = x√(9/4)
- BR = x√(9/2)
- CR = x√(3/2)
- DR = x√(3/4)
10.A cuboid measures 5 cm by 7 cm by 9.5 cm. Work out its surface area.
Medium
- A298 cm²
- B332.5 cm²
- C343 cm²
- D311 cm²
11.What is the formula for the volume of a prism with cross-sectional area A and length l?
Easy
- AV = Al
- BV = A + l
- CV = A/l
- DV = l/A
12.Water flows at 20 cm/s along a rectangular channel of width 15 cm and depth 2.5 cm. How much water flows into the lake in 1 hour? Give your answer in litres.
Hard
- A2700 litres
- B270 litres
- C27000 litres
- D27 litres
13.A solid metal cuboid of volume 600 cm³ is melted and made into 1120 spheres each of radius 0.45 cm. Find the volume of metal not used. (Volume of sphere = (4/3)πr³)
Hard
- A172 cm³
- B428 cm³
- C600 cm³
- D0 cm³
14.A cone has base radius 5x cm and height 12x cm. Its total surface area equals that of a sphere of radius r cm. Show that r² = (45/2)x². Which step is correct?
Hard
- ASlant height l = 13x, cone total SA = π(5x)(13x) + π(5x)² = 65πx² + 25πx² = 90πx². Sphere SA = 4πr². Equate: 4πr² = 90πx² => r² = (90/4)x² = (45/2)x².
- BSlant height l = √((5x)²+(12x)²)=13x, cone curved SA = π(5x)(13x)=65πx², base area = 25πx², total = 90πx². Sphere SA = 4πr². So r² = (90/4)x² = 22.5x² = (45/2)x².
- CCone total SA = πrl + πr² = π(5x)(12x) + π(5x)² = 60πx² + 25πx² = 85πx², then r² = 85/4 x².
- DCone total SA = πr(r+l) = 5πx(5x+13x)=5πx(18x)=90πx², sphere SA = 4πr², so r² = (90/4)x² = (45/2)x².
15.A cuboid has length 6 cm, width 4 cm, and height 3 cm. What is its volume?
Easy
- A72 cm³
- B13 cm³
- C52 cm³
- D36 cm³
16.A cube has side length 5 cm. What is its volume?
Easy
- A25 cm³
- B125 cm³
- C15 cm³
- D100 cm³
17.A cylinder has radius 4 cm and height 10 cm. Calculate its volume in terms of π.
Easy
- A40π cm³
- B160π cm³
- C80π cm³
- D20π cm³
18.A rectangular prism has base area 15 cm² and length 8 cm. What is its volume?
Easy
- A120 cm³
- B23 cm³
- C60 cm³
- D150 cm³
19.The volume of a cuboid is 240 cm³. Its base is a square of side 4 cm. What is its height?
Easy
- A15 cm
- B60 cm
- C12 cm
- D20 cm
20.A cylinder has radius r and height 5r. Its volume is 320π cm³. Find r.
Medium
- A4 cm
- B8 cm
- C5 cm
- D6 cm
21.A sphere has radius 3 cm. What is its volume in terms of π?
Easy
- A36π cm³
- B12π cm³
- C27π cm³
- D9π cm³
22.A cuboid measures 2 cm by 3 cm by 4 cm. What is its surface area?
Easy
- A52 cm²
- B24 cm²
- C26 cm²
- D48 cm²
23.A cube has side length 6 cm. What is its total surface area?
Easy
- A216 cm²
- B36 cm²
- C72 cm²
- D144 cm²
24.A cylinder has radius 3 cm and height 7 cm. Calculate its total surface area in terms of π.
Medium
- A60π cm²
- B42π cm²
- C21π cm²
- D84π cm²
25.A sphere has surface area 144π cm². What is its radius?
Medium
- A6 cm
- B12 cm
- C4 cm
- D8 cm
26.A cone has base radius 5 cm and slant height 13 cm. What is its total surface area in terms of π?
Medium
- A90π cm²
- B65π cm²
- C115π cm²
- D25π cm²
27.A solid cylinder has radius x cm and height 4x cm. Its total surface area is 90π cm². Find x.
Hard
- A3 cm
- B5 cm
- C6 cm
- D2 cm
28.The diagram shows a cuboid. AB = 8 cm, BC = 5 cm, CG = 11 cm. Work out its volume.
Easy
- A440 cm³
- B400 cm³
- C880 cm³
- D220 cm³
29.The diagram shows a prism with cross-sectional area 30 cm² and length 25 cm. What is its volume?
Easy
- A750 cm³
- B55 cm³
- C5 cm³
- D150 cm³
30.The volume of a cuboid is 180 cm³ and its base is a square of side 6 cm. What is its height?
Medium
- A5 cm
- B30 cm
- C3 cm
- D15 cm
31.A water tank is a cuboid of length 1.5 m and width 1 m. The water depth is 60 cm. How many litres of water are in the tank? (1 litre = 1000 cm³)
Medium
- A900 litres
- B90 litres
- C9000 litres
- D9 litres
32.What is the formula for the volume of a cylinder with radius r and height h?
Easy
- AV = π r² h
- BV = 2 π r h
- CV = π r h
- DV = (1/3) π r² h
33.The volume of a prism with cross-sectional area 12 cm² and length 5 cm is:
Easy
- A60 cm³
- B17 cm³
- C7 cm³
- D24 cm³
34.The volume of a pyramid is one-third of the volume of a prism with the same base area and height.
Easy
True or false?
35.A cone has base radius 4 cm and height 9 cm. What is its volume?
Medium
- A48π cm³
- B36π cm³
- C144π cm³
- D12π cm³
36.Which of the following formulas are correct for surface area? (Select all that apply)
Medium
- ACurved surface area of a cylinder: 2πrh
- BTotal surface area of a cylinder: 2πrh + 2πr²
- CCurved surface area of a cone: πrl
- DTotal surface area of a sphere: 2πr²
- ESurface area of a hemisphere (including base): 3πr²
37.Match each 3D shape to its correct volume formula.
Medium
- Cuboid
- Prism
- Cylinder
- Cone
- V = lwh
- V = Al
- V = πr²h
- V = (1/3)πr²h
38.Arrange the following steps to calculate the volume of a cylinder with radius 5 cm and height 10 cm. (Use π = 3.14)
Medium
- Multiply by height: 78.5 × 10 = 785
- Square the radius: 5² = 25
- Multiply by π: 25 × 3.14 = 78.5
- Write formula: V = πr²h
39.Which of the following are correct formulas for surface area? (Select all that apply)
Medium
- ACurved surface area of a cylinder: 2πrh
- BSurface area of a sphere: 4πr²
- CTotal surface area of a cone: πrl + πr²
- DSurface area of a cube: 6s²
- ECurved surface area of a cone: πr²
40.A cylinder has radius 5 cm and height 12 cm. Calculate its total surface area in terms of π.
Medium
- A120π cm²
- B170π cm²
- C100π cm²
- D60π cm²
41.A cube has side length 4 cm. What is its volume in cm³?
Easy
42.Complete the sentence.
Easy
The surface area of a sphere with radius r is given by the formula ____.
43.A cylinder has radius 3 cm and height 10 cm. Calculate its volume in terms of π.
Easy
44.A sphere has radius 6 cm. Calculate its volume. Give your answer in terms of π.
Easy
45.A hemisphere has radius 6 cm. Calculate its total surface area. Give your answer in terms of π.
Medium
46.A cube has side length 7 cm. What is its volume in cm³?
Easy
47.A solid cylinder has radius x cm and height (7x)/2 cm. The total surface area of the cylinder equals the surface area of a sphere of radius R cm. Find R in terms of x.
Hard
48.A sphere has a volume of 288π cm³. Find its radius.
Medium
49.Write down the formula for the total surface area of a cylinder with radius r and height h.
Easy
50.A cone has base radius 6 cm and height 8 cm. Calculate its volume. Give your answer in terms of π.
Easy
51.A solid cone has base radius 5 cm and slant height 13 cm. Calculate the total surface area of the cone. Give your answer in terms of π.
Medium
52.A cube has side length 4 cm. What is its volume in cm³?
Easy
- A76 cm³
- B70 cm³
- C64 cm³
- D58 cm³
53.A cylinder has radius 3 cm and height 10 cm. Calculate its volume in terms of π.
Easy
- A108 π cm³
- B90 π cm³
- C99 π cm³
- D81 π cm³
54.A sphere has radius 6 cm. Calculate its volume. Give your answer in terms of π.
Easy
- A317 π cm³
- B259 π cm³
- C346 π cm³
- D288 π cm³
55.A hemisphere has radius 6 cm. Calculate its total surface area. Give your answer in terms of π.
Medium
- A130 π cm²
- B108 π cm²
- C97 π cm²
- D119 π cm²
56.A cube has side length 7 cm. What is its volume in cm³?
Easy
- A309 cm³
- B411 cm³
- C343 cm³
- D377 cm³
57.A sphere has a volume of 288π cm³. Find its radius.
Medium
- A6 cm
- B7 cm
- C8 cm
- D5 cm
58.A cone has base radius 6 cm and height 8 cm. Calculate its volume. Give your answer in terms of π.
Easy
- A96 π cm³
- B116 π cm³
- C86 π cm³
- D106 π cm³

複習筆記

## Volume & Surface Area - Quick Revision ### Volume Formulas - **Cuboid**: \(V = l \times w \times h\) (not given) - **Prism**: \(V = A \times l\) where \(A\) = cross-sectional area, \(l\) = length (given) - **Cylinder**: \(V = \pi r^2 h\) (given) - **Pyramid**: \(V = \frac{1}{3} A h\) (given) - **Cone**: \(V = \frac{1}{3} \pi r^2 h\) (given) - **Sphere**: \(V = \frac{4}{3} \pi r^3\) (given) ### Surface Area Formulas - **Cuboid**: Sum of areas of 6 faces (2× each pair) - **Cylinder**: Curved = \(2\pi r h\) (given); Total = \(2\pi r h + 2\pi r^2\) - **Cone**: Curved = \(\pi r l\) (given); Total = \(\pi r l + \pi r^2\) - **Sphere**: \(A = 4\pi r^2\) (given) - **Hemisphere**: Curved = \(2\pi r^2\); Total = \(3\pi r^2\) ### Problem Solving Tips - For compound shapes, break into standard solids and add/subtract volumes. - For frustums, subtract smaller cone/pyramid from larger. - For prisms with compound cross-section, find area of cross-section first. - Watch units: convert cm to m, litres (1 litre = 1000 cm³). - Read carefully: sometimes only curved surface area is asked. ### Key Points - Only cuboid volume formula is not given; memorise it. - All other volume and curved surface formulas are in the exam. - Total surface area formulas for cylinder, cone, hemisphere are not given; derive by adding base areas. - In non-calculator questions, leave answers in terms of π. - For spheres, remember diameter = 2r. ### Common Question Types - Volume of prism given cross-sectional area and length. - Volume of cylinder/cone/sphere (exact or in terms of π). - Surface area of cuboid (net helpful). - Total surface area of cylinder/cone/hemisphere. - Problem solving: water flow, melting solids, cost calculations.

Volume: The amount of space a 3D shape occupies, measured in cubic units.
Surface Area: The total area of all the faces (flat or curved) of a 3D shape.
Prism: A 3D shape with a constant cross-section along its length; volume = cross-sectional area × length.
Cylinder: A prism with circular cross-section; volume = πr²h, curved surface area = 2πrh.
Cone: A pyramid with a circular base; volume = (1/3)πr²h, curved surface area = πrl (l = slant height).
Sphere: A perfectly round 3D shape; volume = (4/3)πr³, surface area = 4πr².
Hemisphere: Half a sphere; curved surface area = 2πr², total surface area = 3πr².
Frustum: A truncated cone or pyramid; volume = volume of larger cone/pyramid minus volume of smaller.
Cross-section: The shape obtained by cutting through a 3D object perpendicular to its length; used to find volume of prisms.
Slant Height: The distance along the curved surface from the apex to the base edge of a cone (l).

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