3D shapes, volume and surface area

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Faces, Edges and Vertices

A face is a flat surface of a 3D shape.
An edge is a line where two faces meet.
A vertex (plural: vertices) is a point where edges meet.
For a cuboid: 6 faces, 12 edges, 8 vertices.
For a cube: 6 faces, 12 edges, 8 vertices.
For a cylinder: 3 faces (2 circular, 1 curved), 2 edges, 0 vertices.
For a sphere: 1 curved face, 0 edges, 0 vertices.
For a square-based pyramid: 5 faces, 8 edges, 5 vertices.

Nets of 3D Shapes

A net is a 2D shape that folds up to make a 3D shape.
A cube net is made of 6 squares.
A cuboid net is made of 6 rectangles (3 pairs of identical rectangles).
A cylinder net is made of 2 circles and 1 rectangle.
A square-based pyramid net is made of 1 square and 4 triangles.
A triangular prism net is made of 2 triangles and 3 rectangles.

Plans and Elevations

A plan is the view from above (bird's-eye view).
A front elevation is the view from the front.
A side elevation is the view from the side.
Plans and elevations are 2D drawings that show the shape of a 3D object from different directions.
They are used in design and architecture to describe 3D objects accurately.

Volume of Cubes and Cuboids

Volume is the amount of space inside a 3D shape.
Volume is measured in cubic units, e.g. cm³, m³.
Volume of a cube: V = s³, where s = side length.
Volume of a cuboid: V = l × w × h, where l = length, w = width, h = height.
Example: a cuboid with l = 5 cm, w = 3 cm, h = 2 cm has V = 5 × 3 × 2 = 30 cm³.

Volume of Prisms

A prism is a 3D shape with the same cross-section all along its length.
Volume of a prism: V = A × l, where A = area of cross-section, l = length.
The cross-section is the shape you see when you slice the prism parallel to its base.
Example: a triangular prism with cross-section area 12 cm² and length 10 cm has V = 12 × 10 = 120 cm³.
Cubes and cuboids are special types of prism.

Volume of Cylinders

A cylinder is a prism with a circular cross-section.
Volume of a cylinder: V = π r² h, where r = radius, h = height.
Example: a cylinder with r = 3 cm and h = 10 cm has V = π × 3² × 10 = 90π cm³ ≈ 282.7 cm³.
Use π ≈ 3.14 or the π button on your calculator.

Surface Area of Cuboids

Surface area is the total area of all the faces of a 3D shape.
Surface area is measured in square units, e.g. cm², m².
A cuboid has 6 rectangular faces; opposite faces are identical.
Surface area of a cuboid: SA = 2(lw + lh + wh), where l = length, w = width, h = height.
Example: a cuboid with l = 5 cm, w = 3 cm, h = 2 cm has SA = 2(5×3 + 5×2 + 3×2) = 2(15 + 10 + 6) = 62 cm².

Surface Area of Prisms

To find the surface area of a prism, add the areas of all its faces.
For a prism, the two end faces are identical (the cross-section).
The other faces are rectangles; their total area is perimeter of cross-section × length.
Surface area of a prism: SA = 2A + P × l, where A = area of cross-section, P = perimeter of cross-section, l = length.
Example: a triangular prism with cross-section area 12 cm², perimeter 16 cm, length 10 cm has SA = 2×12 + 16×10 = 24 + 160 = 184 cm².

Surface Area of Cylinders

A cylinder has 2 circular faces and 1 curved surface.
The curved surface unfolds into a rectangle with width = circumference of the circle (2πr) and height = h.
Surface area of a cylinder: SA = 2πr² + 2πrh, where r = radius, h = height.
Example: a cylinder with r = 3 cm and h = 10 cm has SA = 2π×3² + 2π×3×10 = 18π + 60π = 78π cm² ≈ 245.0 cm².

Capacity and Litres

Capacity is the amount a container can hold.
Capacity is often measured in litres (L) or millilitres (mL).
1 litre = 1000 millilitres.
1 litre = 1000 cm³.
1 millilitre = 1 cm³.
1 m³ = 1000 litres.
To convert from cm³ to litres, divide by 1000.
To convert from litres to cm³, multiply by 1000.

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Câu hỏi luyện tập

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1.Which of these is an SI derived unit of volume?
Easy
- Acubic metre
- Bkilogram
- Cnewton
- Dmetre
2.The volume of a container is generally understood to be its capacity. Which statement describes capacity?
Easy
- AThe amount of fluid the container could hold
- BThe amount of space the container itself displaces
- CThe total length of all its edges
- DThe total area of all its faces
3.A cube has edge length 1 dm. What is its volume in litres?
Easy
- A1 litre
- B10 litres
- C0.1 litres
- D1000 litres
4.A cuboid has length 5 cm, width 4 cm and height 3 cm. What is its volume?
Medium
- A60 cm³
- B12 cm³
- C47 cm³
- D94 cm³
5.A prism has a cross-sectional area of 12 cm² and a length of 9 cm. What is its volume?
Medium
- A108 cm³
- B21 cm³
- C54 cm³
- D216 cm³
6.A cylinder has radius 3 cm and height 10 cm. Using π = 3.14, what is its volume to 1 decimal place?
Medium
- A282.6 cm³
- B94.2 cm³
- C188.4 cm³
- D565.2 cm³
7.A cuboid has length 8 cm, width 5 cm and height 2 cm. What is its total surface area?
Medium
- A132 cm²
- B80 cm²
- C66 cm²
- D264 cm²
8.A cylinder has radius 4 cm and height 7 cm. Using π = 3.14, what is its total surface area to 1 decimal place?
Hard
- A276.3 cm²
- B175.8 cm²
- C351.7 cm²
- D100.5 cm²

Thêm câu hỏi

9.A cuboid has 6 faces. Which of these statements are true? (select all that apply)
Medium
- AA cuboid has 12 edges.
- BA cuboid has 8 vertices.
- CAll 6 faces of a cuboid must be squares.
- DA cuboid has 6 vertices.
10.The volume of a container is generally understood to be its capacity.
Easy
True or false?
11.Which of the following statements about volume are true? (select all that apply)
Medium
- AVolume measures regions in three-dimensional space.
- BVolume can be measured in cubic metres or litres.
- CZero-dimensional objects have a volume of 0.
- DOne-dimensional objects have a volume of 1.
- EIn four dimensions, an analogous concept is called hypervolume.
12.Match each 3D shape with the formula for its volume.
Medium
- Cube with edge a
- Cuboid l × w × h
- Cylinder radius r, height h
- Prism cross-section A, length L
- V = a³
- V = l × w × h
- V = πr²h
- V = A × L
13.Order these units of volume from smallest to largest.
Easy
- 1 cm³
- 1 litre
- 1 m³
- 1 dm³
14.A cuboid has length 10 cm, width 4 cm and height 2 cm. Which of the following are true? (select all that apply)
Medium
- AIts volume is 80 cm³.
- BIts total surface area is 136 cm².
- CIts volume is 40 cm³.
- DIts total surface area is 80 cm².
- EThe area of its front face is 20 cm².
15.A cylinder has radius 5 cm and height 8 cm. Using π = 3.14, what is its volume to 1 decimal place?
Medium
- A628.0 cm³
- B314.0 cm³
- C1256.0 cm³
- D251.2 cm³
16.The volume of a cylinder is given by the formula V = 2πrh.
Medium
True or false?
17.A cuboid has length 6 cm, width 4 cm and height 2 cm. What is its total surface area?
Medium
- A88 cm²
- B48 cm²
- C44 cm²
- D96 cm²
18.Which of the following are units of volume? (select all that apply)
Medium
- ACubic metre
- BLitre
- CSquare metre
- DGallon
- EMetre
19.A cuboid measures 8 cm by 5 cm by 2 cm. Work out its volume.
Medium
- A80 cm³
- B160 cm³
- C40 cm³
- D26 cm³
20.A prism has a cross-sectional area of 24 cm² and a length of 5 cm. What is its volume?
Medium
- A120 cm³
- B60 cm³
- C29 cm³
- D48 cm³
21.A cylinder has radius 3 cm and height 10 cm. Using π = 3.14, work out its curved surface area (the area of the curved side only).
Medium
- A188.4 cm²
- B94.2 cm²
- C282.6 cm²
- D60.0 cm²
22.A cuboid has length 6 cm, width 4 cm and height 3 cm. Which of the following statements are true? (select all that apply)
Medium
- AIts volume is 72 cm³.
- BIts total surface area is 108 cm².
- CIts volume is 13 cm³.
- DIts total surface area is 54 cm².
23.Order these volumes from smallest to largest.
Medium
- 1 cm³
- 1 litre
- 1 m³
- 1 dm³
24.A water tank in the shape of a cuboid has length 1.2 m, width 0.5 m and height 0.8 m. It is filled to a depth of 0.6 m. Using 1 m³ = 1000 litres, how many litres of water are in the tank?
Medium
- A360 litres
- B480 litres
- C576 litres
- D360000 litres
25.A solid metal prism has a right-angled triangular cross-section with base 6 cm and perpendicular height 8 cm. The prism is 20 cm long. The metal has a mass of 7.8 g per cm³. What is the mass of the prism?
Hard
- A3744 g
- B7488 g
- C1872 g
- D936 g
26.A cylindrical pipe has an outer radius of 4 cm and an inner radius of 3 cm. The pipe is 50 cm long. Taking π = 3.14, what is the volume of metal in the pipe, to 1 decimal place?
Hard
- A1099.0 cm³
- B785.0 cm³
- C314.0 cm³
- D2198.0 cm³
27.A cuboid has volume 120 cm³. Its length is 8 cm and its width is 5 cm. What is its height?
Medium
- A3 cm
- B2 cm
- C4 cm
- D15 cm
28.A cube has total surface area 294 cm². What is the volume of the cube?
Hard
- A343 cm³
- B294 cm³
- C216 cm³
- D512 cm³
29.A cuboid measures 10 cm by 6 cm by 4 cm. A cube has edge length 5 cm. Which solid has the greater volume, and by how much?
Medium
- AThe cuboid, by 115 cm³
- BThe cuboid, by 240 cm³
- CThe cube, by 115 cm³
- DThe cube, by 125 cm³
30.Which of the following statements about a cylinder of radius r and height h are true? (select all that apply)
Medium
- AIts volume is πr²h.
- BIts curved surface area is 2πrh.
- CIts total surface area is 2πrh + 2πr².
- DIts volume is 2πrh.
- EIts total surface area is πr²h.
31.A cuboid has length 12 cm, width 5 cm and height 4 cm. Which of the following statements are true? (select all that apply)
Hard
- AIts volume is 240 cm³.
- BIts total surface area is 256 cm².
- CThe area of one 12 cm by 5 cm face is 60 cm².
- DIts volume is 120 cm³.
- EIts total surface area is 128 cm².
32.A cube of edge length 3 cm and a cuboid measuring 9 cm by 2 cm by 1 cm have the same volume.
Medium
True or false?
33.If the radius of a cylinder is doubled and its height is halved, the volume is doubled.
Medium
True or false?
34.Match each solid with the correct expression for its volume.
Medium
- Cube of edge x
- Cuboid l × w × h
- Cylinder radius r, height h
- x³
- l × w × h
- πr²h
35.Which of these is the SI derived unit of volume?
Easy
- Acubic metre
- Bsquare metre
- Cmetre
- Dkilogram
36.The litre is equal to one cubic decimetre.
Easy
True or false?
37.Which of these is the correct SI derived unit for the volume of a liquid?
Easy
- Alitre
- Bmillilitre
- Ccubic metre
- Dgallon
38.A cuboid has volume 240 cm³. Its cross-sectional area is 30 cm². What is its length?
Medium
- A6 cm
- B8 cm
- C12 cm
- D7200 cm
39.The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height.
Easy
True or false?
40.Match each 3D shape with the number of faces it has.
Medium
- Cube
- Cylinder
- Square-based pyramid
- Triangular prism
- 5
- 6
- 3
- 2
41.Complete the statement about units of volume.
Easy
The litre is defined as 1 ____.
42.A cuboid has length 6 cm, width 2 cm and height 5 cm. Work out its volume in cm³.
Medium
43.A prism has a cross-sectional area of 15 cm² and a length of 12 cm. Calculate its volume in cm³.
Hard
44.Complete the statement about the faces, edges and vertices of a cube.
Easy
A cube has ____ faces, ____ edges and ____ vertices.
45.Calculate the volume of the cuboid in cm³.
Medium
46.Calculate the total surface area of the cuboid in cm².
Medium
47.Which SI derived unit of volume is defined as the volume of a cube with edges of 1 decimetre?
Easy
48.A cylinder has radius 5 cm and height 6 cm. Using π = 3.14, calculate its volume in cm³, giving your answer to 1 decimal place.
Medium
49.A cuboid has length 9 cm, width 3 cm and height 2 cm. Calculate its total surface area in cm².
Medium
50.Complete the statement about the units used for a cuboid's volume.
Medium
The volume of a cuboid with edges measured in centimetres is measured in ____.
51.A cuboid has length 7 cm, width 3 cm and height 4 cm. Work out its volume in cm³.
Easy
52.A prism has a cross-sectional area of 18 cm² and a length of 6 cm. Calculate its volume in cm³.
Medium
53.Explain how you would find the volume of an irregular solid, such as a small stone, using water.
Medium
54.Describe how you would find the volume of an irregular solid, such as a small stone, using water.
Hard
55.Explain how to find the volume of a cuboid and state the units you would use if the edges are measured in centimetres.
Medium
56.A sealed cuboid box has external dimensions 20 cm by 15 cm by 10 cm and is made of wood 1 cm thick. Explain how to find the volume of the hollow space inside the box.
Hard
57.Explain how the formula for the volume of any prism is related to its cross-section.
Hard
58.A cuboid has length 6 cm, width 2 cm and height 5 cm. Work out its volume in cm³.
Medium
- A60 cm³
- B54 cm³
- C66 cm³
- D72 cm³
59.A prism has a cross-sectional area of 15 cm² and a length of 12 cm. Calculate its volume in cm³.
Hard
- A216 cm³
- B162 cm³
- C180 cm³
- D198 cm³
60.Calculate the volume of the cuboid in cm³.
Medium
- A144 cm³
- B108 cm³
- C120 cm³
- D132 cm³
61.Calculate the total surface area of the cuboid in cm².
Medium
- A174 cm²
- B158 cm²
- C142 cm²
- D190 cm²
62.A cylinder has radius 5 cm and height 6 cm. Using π = 3.14, calculate its volume in cm³, giving your answer to 1 decimal place.
Medium
- A518 cm³
- B424 cm³
- C471 cm³
- D565 cm³
63.A cuboid has length 9 cm, width 3 cm and height 2 cm. Calculate its total surface area in cm².
Medium
- A92 cm²
- B112 cm²
- C102 cm²
- D122 cm²
64.A cuboid has length 7 cm, width 3 cm and height 4 cm. Work out its volume in cm³.
Easy
- A92 cm³
- B100 cm³
- C76 cm³
- D84 cm³
65.A prism has a cross-sectional area of 18 cm² and a length of 6 cm. Calculate its volume in cm³.
Medium
- A130 cm³
- B108 cm³
- C119 cm³
- D97 cm³

Ghi chú ôn tập

## 3D Shapes: Volume & Surface Area ### Faces, Edges, Vertices - **Face**: flat surface - **Edge**: line where two faces meet - **Vertex** (plural vertices): corner where edges meet ### Nets, Plans & Elevations - A **net** is a 2D shape that folds to make a 3D shape. - **Plan** = view from above; **elevations** = views from the front and side. ### Volume of Prisms A **prism** has the same cross-section all along its length. Volume = area of cross-section × length - **Cube**: V = s^3 (s = side length) - **Cuboid**: V = l × w × h (l = length, w = width, h = height) - **Cylinder**: V = π r^2 h (r = radius, h = height) ### Surface Area - **Cuboid**: SA = 2(lw + lh + wh) - **Cylinder**: SA = 2π r^2 + 2π r h - 2π r^2 = two circular ends - 2π r h = curved surface - For any prism: SA = (perimeter of cross-section × length) + 2 × area of cross-section ### Capacity & Litres - Capacity = amount a container can hold. - 1 litre = 1000 cm^3 - 1 m^3 = 1000 litres ### Units - Volume uses cubic units: cm^3, m^3 - 1 m^3 = 1 000 000 cm^3 ### Quick Tips - Always check units match before calculating. - For cylinders, use π ≈ 3.14 unless told otherwise. - Surface area is measured in square units (cm^2, m^2).

Face: A flat surface of a 3D shape.
Edge: A line where two faces meet.
Vertex: A corner where edges meet (plural: vertices).
Net: A 2D shape that can be folded to make a 3D shape.
Plan: The view of a 3D shape from above.
Elevation: A view of a 3D shape from the front or side.
Prism: A 3D shape with the same cross-section all along its length.
Volume: The amount of space a 3D shape occupies, measured in cubic units.
Surface area: The total area of all the faces of a 3D shape, measured in square units.
Capacity: The amount of fluid a container can hold.
Litre: A unit of capacity; 1 litre = 1000 cm^3.
Cylinder: A prism with a circular cross-section.

Thẻ ghi nhớ

What is a face of a 3D shape?

A face is a flat surface of a 3D shape. For example, a cube has 6 faces.

What is an edge of a 3D shape?

An edge is a line where two faces meet. For example, a cube has 12 edges.

What is a vertex of a 3D shape?

A vertex is a point where edges meet (a corner). For example, a cube has 8 vertices.

What is a net of a 3D shape?

A net is a flat pattern that can be folded to make the 3D shape. For example, a cube net is made of 6 squares.

What are plans and elevations?

Plans and elevations are 2D views of a 3D shape: the plan is the view from above, and elevations are views from the front and side.

How do you find the volume of a cube with side length s?

Volume of a cube = s³ (s = side length). For example, if s = 3 cm, volume = 27 cm³.

How do you find the volume of a cuboid with length l, width w, and height h?

Volume of a cuboid = l × w × h (l = length, w = width, h = height). For example, 2 cm × 3 cm × 4 cm = 24 cm³.

How do you find the volume of any prism?

Volume of a prism = area of cross-section × length. For example, a triangular prism with cross-section area 10 cm² and length 5 cm has volume 50 cm³.

How do you find the volume of a cylinder with radius r and height h?

Volume of a cylinder = π × r² × h (r = radius, h = height). For example, r = 2 cm, h = 5 cm gives volume = 20π cm³.

How do you find the surface area of a cube with side length s?

Surface area of a cube = 6 × s² (s = side length). For example, if s = 2 cm, surface area = 24 cm².

How do you find the surface area of a cuboid with length l, width w, and height h?

Surface area of a cuboid = 2 × (lw + lh + wh) (l = length, w = width, h = height). For example, 2 × (2×3 + 2×4 + 3×4) = 52 cm².

How do you find the surface area of a cylinder with radius r and height h?

Surface area of a cylinder = 2 × π × r² + 2 × π × r × h (r = radius, h = height). This includes the two circular ends and the curved surface.

What is the relationship between volume and capacity?

The volume of a container is its capacity: the amount of fluid it can hold. For example, a 1-litre bottle has a capacity of 1 litre.

How many cubic centimetres are in 1 litre?

1 litre = 1000 cm³. This is because 1 litre is the volume of a cube with sides 10 cm.

How many litres are in 1 cubic metre?

1 m³ = 1000 litres. This is because 1 m³ = 1,000,000 cm³ and 1 litre = 1000 cm³.

What is the formula for the volume of a prism?

Volume = area of cross-section × length. This works for any prism, including cuboids and cylinders.

What is the formula for the surface area of a prism?

Surface area = sum of the areas of all faces. For a prism, this is 2 × area of cross-section + perimeter of cross-section × length.

What is the formula for the volume of a cylinder?

Volume = π × r² × h (r = radius, h = height). For example, r = 3 cm, h = 7 cm gives volume = 63π cm³.

What is the formula for the surface area of a cylinder?

Surface area = 2πr² + 2πrh (r = radius, h = height). The first term is the area of the two circular ends, the second is the curved surface.

What is a prism?

A prism is a 3D shape with a constant cross-section along its length. Examples include cuboids, triangular prisms, and cylinders.

How do you find the volume of an irregular object?

Submerge it in water and measure the volume of water displaced. The volume of the object equals the difference in water level.

What is the volume of a shape with a cross-sectional area of 15 cm² and a length of 4 cm?

Volume = area of cross-section × length = 15 cm² × 4 cm = 60 cm³.

What is the surface area of a cube with side length 5 cm?

Surface area = 6 × s² = 6 × 5² = 6 × 25 = 150 cm².

What is the volume of a cylinder with radius 2 cm and height 10 cm?

Volume = π × r² × h = π × 2² × 10 = 40π cm³ (approximately 125.7 cm³).

What is the surface area of a cuboid with length 3 cm, width 4 cm, and height 5 cm?

Surface area = 2 × (lw + lh + wh) = 2 × (3×4 + 3×5 + 4×5) = 2 × (12 + 15 + 20) = 2 × 47 = 94 cm².

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