Volume And Surface Area

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课程笔记

Volume of Cubes and Cuboids

Volume is the amount of space a 3D shape takes up.
For a cuboid with length l, width w, height h: V = lwh.
A cube is a special cuboid with all sides equal: V = s³.
Units are cubic, e.g. cm³, m³.

Cuboid

Volume of Prisms

A prism has a constant cross-sectional area A and length l.
Volume formula: V = A l (given in exam).
The cross-section can be any shape (e.g. triangle, L-shape).
Find the area of the cross-section first, then multiply by length.

Prism

Volume of Cylinders

A cylinder is like a prism with a circular base.
Volume: V = \π r² h (given in exam).
r = radius, h = height.

Cylinder

Volume of Pyramids and Cones

Pyramid volume: V = \frac{1}{3} A h (given), where A = base area, h = perpendicular height.
Cone volume: V = \frac{1}{3} \π r² h (given).
The height must be perpendicular to the base.

Pyramid

Volume of Spheres

Sphere volume: V = \frac{4}{3} \π r³ (given).
A hemisphere is half a sphere: V = \frac{2}{3} \π r³.

Sphere

Problem Solving with Volumes

Real-life problems often combine volume with other topics (e.g. money).
For compound shapes, split into standard solids and add volumes.
For fractions of shapes (e.g. hemisphere), find the full volume then take the fraction.
Always check units and round as required (e.g. 3 significant figures).

Compound L-shaped prism

Surface Area of Cuboids and Prisms

Surface area is the sum of areas of all faces.
For a cuboid: SA = 2(lw + lh + wh).
Drawing a net helps identify all faces.
For prisms, calculate area of each face (including bases) and add.

Cuboid

Surface Area of Cylinders

Curved surface area: A = 2\π r h (given).
Total surface area: SA = 2\π r h + 2\π r² (not given).
The net is two circles and a rectangle.

Net of a cylinder

Surface Area of Cones and Spheres

Cone curved surface: A = \π r l (given), l = slant height.
Cone total: SA = \π r l + \π r².
Sphere surface: A = 4\π r² (given).
Hemisphere: SA = 2\π r² + \π r² = 3\π r².

Surface area of a hemisphere

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练习题

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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 = 2lw + 2lh + 2wh
- CV = l + w + h
- DV = l² + w² + h²
2.A cube has side length 8 cm. What is its surface area?
Easy
- A384 cm²
- B64 cm²
- C512 cm²
- D192 cm²
3.A cuboid measures 5 cm by 7 cm by 9.5 cm. What is its surface area?
Medium
- A298 cm²
- B346 cm²
- C270.75 cm²
- D332.5 cm²
4.A cylinder has radius 6 cm and height 17 cm. What is its volume? (Use π ≈ 3.142 or calculator)
Medium
- A1923 cm³
- B1922 cm³
- C1924 cm³
- D1920 cm³
5.A cone has radius 4.5 cm and height 10.4 cm. What is its volume in terms of π?
Medium
- A70.2π cm³
- B70.2π cm²
- C210.6π cm³
- D23.4π cm³
6.A cube has a volume of 1000 cm³. What is its surface area?
Medium
- A600 cm²
- B100 cm²
- C500 cm²
- D400 cm²
7.A sphere has radius 5.2 cm. What is its surface area? (Use π ≈ 3.142)
Medium
- A339.8 cm²
- B339.8 cm³
- C340 cm²
- D339 cm²
8.The volume of a cuboid is 180 cm³. Its base is a square of side 6 cm. What is its height?
Medium
- A5 cm
- B30 cm
- C6 cm
- D3 cm

复习笔记

## Volume & Surface Area - Quick Revision ### Volume Formulas - **Cuboid:** V = lwh (length × width × height) - **Prism:** V = Al (cross-sectional area × length) — given in exam - **Cylinder:** V = πr²h — given in exam - **Pyramid:** V = ⅓Ah (base area × perpendicular height ÷ 3) — given in exam - **Cone:** V = ⅓πr²h — given in exam - **Sphere:** V = ⁴⁄₃πr³ — given in exam ### Surface Area Formulas - **Cuboid:** Sum of areas of all 6 faces (2lw + 2lh + 2wh) - **Cylinder:** Curved = 2πrh (given); Total = 2πrh + 2πr² - **Cone:** Curved = πrl (given); Total = πrl + πr² - **Sphere:** A = 4πr² (given) - **Hemisphere:** Curved = 2πr²; Total = 3πr² ### Key Points - For prisms, volume = area of cross-section × length. - For compound shapes, break into standard solids and add volumes. - Always check if you need total surface area or just curved/lateral area. - Use nets to help visualise faces for surface area. - Read questions carefully: sometimes you need to find a missing dimension (e.g., height from volume). ### Common Mistakes - Forgetting to include all faces in surface area. - Using diameter instead of radius in circle formulas. - Mixing up slant height (l) and perpendicular height (h) in cones. - Not converting units if required. ### Tips from Past Papers - Easy questions often involve cuboid volume/surface area. - Medium questions include cylinders, cones, spheres, and problem-solving with volumes. - Hard questions may combine shapes or require working backwards from volume to find a dimension.

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 running through it; volume = area of cross-section × length.
Cylinder: A prism with circular cross-section; volume = πr²h, curved surface area = 2πrh.
Cone: A 3D shape with a circular base and a vertex; volume = ⅓πr²h, curved surface area = πrl (l = slant height).
Sphere: A perfectly round 3D shape; volume = ⁴⁄₃πr³, surface area = 4πr².
Cuboid: A rectangular-based prism; volume = lwh, surface area = 2lw + 2lh + 2wh.
Net: A 2D pattern that can be folded to form a 3D shape, useful for calculating surface area.
Slant Height: The distance along the curved surface from the base edge to the vertex of a cone (or pyramid).
Cross-section: The shape obtained by cutting through a 3D object perpendicular to its length; used to find volume of prisms.

闪卡

What is the formula for the volume of a cuboid?

V = lwh, where l is length, w is width, h is height.

What is the formula for the volume of a prism?

V = Al, where A is the cross-sectional area and l is the length.

What is the formula for the volume of a cylinder?

V = πr²h, where r is the radius and h is the height.

What is the formula for the volume of a pyramid?

V = (1/3)Ah, where A is the base area and h is the perpendicular height.

What is the formula for the volume of a cone?

V = (1/3)πr²h, where r is the base radius and h is the perpendicular height.

What is the formula for the volume of a sphere?

V = (4/3)πr³, where r is the radius.

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

A = 2πrh, where r is the radius and h is the height.

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

A = 2πrh + 2πr².

What is the formula for the curved surface area of a cone?

A = πrl, where r is the radius and l is the slant height.

What is the formula for the total surface area of a cone?

A = πrl + πr².

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

A = 4πr², where r is the radius.

What is the surface area of a hemisphere?

A = 2πr² + πr² = 3πr² (curved surface plus base).

A cube has edge length 8 cm. Calculate its surface area.

Surface area = 6 × (8²) = 6 × 64 = 384 cm².

A cuboid measures 15 cm by 12 cm by 4 cm. Calculate its surface area.

Surface area = 2(15×12 + 15×4 + 12×4) = 2(180 + 60 + 48) = 2×288 = 576 cm².

A cuboid measures 5 cm by 7 cm by 9.5 cm. Work out its surface area.

Surface area = 2(5×7 + 5×9.5 + 7×9.5) = 2(35 + 47.5 + 66.5) = 2×149 = 298 cm².

A closed box is 5 cm by 4 cm by 2 cm. Calculate its volume.

Volume = 5 × 4 × 2 = 40 cm³.

A cube has surface area 384 cm². Find the length of one side.

Let side = s. 6s² = 384 → s² = 64 → s = 8 cm.

A cube has volume 1000 cm³. Calculate its surface area.

Side = ∛1000 = 10 cm. Surface area = 6 × 10² = 600 cm².

A cylinder has radius 6 cm and height 17 cm. Show its volume is 1923 cm³ (to 4 s.f.).

V = π × 6² × 17 = π × 36 × 17 = 612π ≈ 1922.65... ≈ 1923 cm³.

A cone has radius 4.5 cm and height 10.4 cm. Calculate its volume in terms of π.

V = (1/3)π × (4.5)² × 10.4 = (1/3)π × 20.25 × 10.4 = (1/3)π × 210.6 = 70.2π cm³.

A sphere has radius 5.2 cm. Work out its surface area.

A = 4π × (5.2)² = 4π × 27.04 = 108.16π ≈ 339.8 cm² (to 1 d.p.).

A sphere has volume 381 cm³. Show its radius is 4.50 cm (to 3 s.f.).

V = (4/3)πr³ = 381 → r³ = (381×3)/(4π) ≈ 90.99 → r ≈ 4.50 cm.

A cuboid has volume 180 cm³ and a square base of side 6 cm. Find its height.

Volume = base area × height → 180 = 36 × h → h = 5 cm.

A cylinder has radius 3 cm and height 4.5 cm. Calculate its total surface area.

Total SA = 2π×3×4.5 + 2π×3² = 27π + 18π = 45π ≈ 141.4 cm².

A plant pot is a cylinder with radius 15 cm and height 24 cm. Calculate its volume.

V = π × 15² × 24 = π × 225 × 24 = 5400π ≈ 16965 cm³.

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