Volume And Surface Area

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Notes

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: $V = s^3$ .
Units are cubic, e.g. $cm^{3}, m^{3}$ .

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.

Volume of Cylinders

A cylinder is like a prism with a circular base.
Volume: ** $V = \pi r^2 h$ ** (given in exam).
$r$ = radius, $h$ = height.

Volume of Pyramids and Cones

Pyramid volume: ** $V = \frac{1}{3} A h$ ** (given), where $A$ = base area, $h$ = perpendicular height.
Cone volume: ** $V = \frac{1}{3} \pi r^2 h$ ** (given).
The height must be perpendicular to the base.

Volume of Spheres

Sphere volume: ** $V = \frac{4}{3} \pi r^3$ ** (given).
A **hemisphere** is half a sphere: $V = \frac{2}{3} \pi r^3$ .

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).

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.

Surface Area of Cylinders

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

Surface Area of Cones and Spheres

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

Cuboid with dimensions l, w, h

Cylinder with radius r and height h

Net of a cuboid

Sphere with radius r

Practice questions

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1.What is the formula for the volume of a cuboid with length l, width w, and height h?
Easy
- A $V = lwh$
- B $V = 2lw + 2lh + 2wh$
- C $V = l + w + h$
- D $V = l^{2} + w^{2} + h^{2}$
2.A cube has side length 8 cm. What is its surface area?
Easy
- A $384 cm^{2}$
- B $64 cm^{2}$
- C $512 cm^{2}$
- D $192 cm^{2}$
3.A cuboid measures 5 cm by 7 cm by 9.5 cm. What is its surface area?
Easy
- A $346 cm^{2}$
- B $332.5 cm^{2}$
- C $295 cm^{2}$
- D $380 cm^{2}$
4.A cylinder has radius 6 cm and height 17 cm. What is its volume? (Use π ≈ 3.142 or calculator)
Medium
- A $1923 cm^{3}$
- B $1922 cm^{3}$
- C $1924 cm^{3}$
- D $1920 cm^{3}$
5.A cone has radius 4.5 cm and height 10.4 cm. What is its volume in terms of π?
Medium
- A $70.2\pi cm^{3}$
- B $70.2\pi cm^{2}$
- C $210.6\pi cm^{3}$
- D $23.4\pi cm^{3}$
6.A cube has a volume of $1000 cm^{3}$ . What is its surface area?
Hard
- A $600 cm^{2}$
- B $100 cm^{2}$
- C $500 cm^{2}$
- D $400 cm^{2}$
7.A sphere has radius 5.2 cm. What is its surface area? (Use π ≈ 3.142)
Medium
- A $339.8 cm^{2}$
- B $339.8 cm^{3}$
- C $340 cm^{2}$
- D $339 cm^{2}$
8.The volume of a cuboid is $180 cm^{3}$ . Its base is a square of side 6 cm. What is its height?
Easy
- A5 cm
- B30 cm
- C6 cm
- D3 cm

Summary

## 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.

Key terms

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.

Slides

Volume of a Cuboid

Volume measures how much space a 3D shape takes up.
For a cuboid, use $V = l \times w \times h (length \times$ width $\times height)$ .
This formula is not given in the exam – memorise it!

A cuboid has length 5 cm, width 4 cm, height 2 cm. What is its volume?

$20 cm^{3}$
$40 cm^{3}$
$11 cm^{3}$
$30 cm^{3}$

Volume of a Prism

A prism has a constant cross-sectional area.
Volume $= cross-sectional$ area × length $(V = Al)$ .
This formula is given in the exam.

Volume of a prism = ____ × length.

Volume of a Cylinder

A cylinder is like a prism with circular cross-section.
Volume $= \pi r^{2}h (given$ in exam).
r is radius, h is height.

A cylinder has radius 8 cm and height 20 cm. Which calculation gives its volume?

$\pi \times 8 \times 20$
$\pi \times 8^{2} \times 20$
$\pi \times 8 \times 20^{2}$
$\pi \times 8^{2} \times 20^{2}$

Volume of a Pyramid and Cone

Pyramid: $V =$ ⅓ × base area × height (given).
Cone: $V =$ ⅓ $\pi r^{2}h (given)$ .
Height must be perpendicular to base.

Which of these are true about volume formulas?

Volume of a cuboid is lwh.
Volume of a cylinder is $\pi r^{2}h$ .
Volume of a cone is $\pi r^{2}h$ .
Volume of a pyramid is ⅓ × base area × height.

Volume of a Sphere

Sphere: $V =$ ⁴⁄₃ $\pi r^{3} (given)$ .
r is the radius.

Volume of a sphere $= ____ \pi r^{3}$ .

Problem Solving with Volumes

Often real-life problems combine shapes (compound objects).
Find volume of each part and add them.
For prisms, find cross-sectional area first.

A prism has cross-sectional area $38 cm^{2}$ and length 10 cm. What is its volume?

$48 cm^{3}$
$380 cm^{3}$
$28 cm^{3}$
$3800 cm^{3}$

Surface Area of Cuboids and Prisms

Surface area is sum of areas of all faces.
For cuboids, calculate area of each rectangle and add.
Drawing a net can help.

A cube has edge length 8 cm. What is its surface area?

$64 cm^{2}$
$384 cm^{2}$
$512 cm^{2}$
$192 cm^{2}$

Surface Area of a Cylinder

Curved surface area $= 2\pi rh (given)$ .
Total surface area $= 2\pi rh + 2\pi r^{2} (not given)$ .
Includes top and bottom circles.

Curved surface area of a cylinder $= 2\pi \times ____ \times h$ .

Surface Area of a Cone and Sphere

Cone curved area $= \pi rl (given)$ , total $= \pi rl + \pi r^{2}$ .
Sphere surface area $= 4\pi r^{2} (given)$ .
Hemisphere: curved half of sphere plus base circle.

Which formulas are given in the exam?

Curved surface area of a cylinder: 2πrh
Total surface area of a cylinder: $2\pi rh + 2\pi r^{2}$
Surface area of a sphere: $4\pi r^{2}$
Curved surface area of a cone: πrl

Worked Example: Toy Surface Area

A toy: cone $(r=5, slant=12)$ on hemisphere $(r=5)$ .
Cone curved area $= \pi \times 5\times 12 = 60\pi$ .
Hemisphere curved area = ½×4π×5² $= 50\pi$ .
Total $= 110\pi$ ≈ $346 cm^{2} (3 s.f.)$ .

A hemisphere has radius 5 cm. What is its curved surface area?

$50\pi cm^{2}$
$100\pi cm^{2}$
$25\pi cm^{2}$
$20\pi cm^{2}$

Flashcards

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?

$V = \pi r^{2}h$ , where r is the radius and h is the height.

What is the formula for the volume of a pyramid?

$V = (\frac{1}{3})Ah$ , where A is the base area and h is the perpendicular height.

What is the formula for the volume of a cone?

$V = (\frac{1}{3})\pi r^{2}h$ , where r is the base radius and h is the perpendicular height.

What is the formula for the volume of a sphere?

$V = (\frac{4}{3})\pi r^{3}$ , where r is the radius.

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

$A = 2\pi rh$ , where r is the radius and h is the height.

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

$A = 2\pi rh + 2\pi r^{2}$ .

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

$A = \pi rl$ , where r is the radius and l is the slant height.

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

$A = \pi rl + \pi r^{2}$ .

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

$A = 4\pi r^{2}$ , where r is the radius.

What is the surface area of a hemisphere?

$A = 2\pi r^{2} + \pi r^{2} = 3\pi r^{2} (curved$ surface plus base).

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

Surface area $= 6 \times (8^{2}) = 6 \times 64 = 384 cm^{2}$ .

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

Surface area $= 2(15\times 12 + 15\times 4 + 12\times 4) = 2(180 + 60 + 48) = 2\times 288 = 576 cm^{2}$ .

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

Surface area $= 2(5\times 7 + 5\times 9.5 + 7\times 9.5) = 2(35 + 47.5 + 66.5) = 2\times 149 = 298 cm^{2}$ .

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

Volume $= 5 \times 4 \times 2 = 40 cm^{3}$ .

A cube has surface area $384 cm^{2}$ . Find the length of one side.

Let side $= s$ . $6s^{2} = 384$ → $s^{2} = 64$ → $s = 8$ cm.

A cube has volume $1000 cm^{3}$ . Calculate its surface area.

Side = ∛1000 $= 10$ cm. Surface area $= 6 \times 10^{2} = 600 cm^{2}$ .

A cylinder has radius 6 cm and height 17 cm. Show its volume is $1923 cm^{3} (to 4 s.f.)$ .

$V = \pi \times 6^{2} \times 17 = \pi \times 36 \times 17 = 612\pi$ ≈ 1922.65... ≈ $1923 cm^{3}$ .

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

$V = (\frac{1}{3})\pi \times (4.5)^{2} \times 10.4 = (\frac{1}{3})\pi \times 20.25 \times 10.4 = (\frac{1}{3})\pi \times 210.6 = 70.2\pi cm^{3}$ .

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

$A = 4\pi \times (5.2)^{2} = 4\pi \times 27.04 = 108.16\pi$ ≈ $339.8 cm^{2} (to 1 d.p.)$ .

A sphere has volume $381 cm^{3}$ . Show its radius is 4.50 cm (to 3 s.f.).

$V = (\frac{4}{3})\pi r^{3} = 381$ → $r^{3} = (381\times 3)/(4\pi )$ ≈ 90.99 → r ≈ 4.50 cm.

A cuboid has volume $180 cm^{3}$ and a square base of side 6 cm. Find its height.

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

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

Total $SA = 2\pi \times 3\times 4.5 + 2\pi \times 3^{2} = 27\pi + 18\pi = 45\pi$ ≈ $141.4 cm^{2}$ .

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

$V = \pi \times 15^{2} \times 24 = \pi \times 225 \times 24 = 5400\pi$ ≈ $16965 cm^{3}$ .

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