Volume And Surface Area

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Notes

Volume of Cubes and Cuboids

Volume of a cuboid: **V = lwh** $(length \times$ width $\times height)$ .
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.

Volume of Prisms

Volume of a prism: **V $= A \times$ 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**.

Volume of Cylinders

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

Volume of Pyramids and Cones

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

Volume of Spheres

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

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.

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.

Surface Area of Cylinders

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

Surface Area of Cones

Curved surface area: **A = πrl** where l is slant height (given in exam).
Total surface area: **A $= \pi rl +$ πr²** (not given).
Slant height l can be found using Pythagoras: **l $= √(r^{2} +$ h²)**.

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**.

Volume of a Cuboid

Volume of a Cylinder

Surface Area of a Cone

Volume of a Sphere

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 $V = l + w + h$
- C $V = 2lw + 2wh + 2lh$
- D $V = l^{2} + w^{2} + h^{2}$
2.The volume of a cube is $1000 cm^{3}$ . What is the side length of the cube?
Easy
- A10 cm
- B100 cm
- C33.3 cm
- D31.6 cm
3.A cuboid has length 8 cm, width 5 cm, and height 11 cm. What is its volume?
Easy
- A $440 cm^{3}$
- B $400 cm^{3}$
- C $352 cm^{3}$
- D $880 cm^{3}$
4.The area of the cross-section of a prism is $30 cm^{2}$ and its length is 25 cm. What is its volume?
Easy
- A $750 cm^{3}$
- B $55 cm^{3}$
- C $150 cm^{3}$
- D $5 cm^{3}$
5.A solid cylinder has radius 3 cm and height 4.5 cm. Calculate its total surface area in terms of π.
Medium
- A $45\pi cm^{2}$
- B $27\pi cm^{2}$
- C $54\pi cm^{2}$
- D $36\pi cm^{2}$
6.The volume of a cuboid is $180 cm^{3}$ and its base is a square of side 6 cm. What is its height?
Medium
- A5 cm
- B30 cm
- C3 cm
- D6 cm
7.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 $= 1000 cm^{3})$
Medium
- A900 litres
- B90 litres
- C9000 litres
- D9 litres
8.A cuboid has base area $7 cm^{2}$ and volume $21 cm^{3}$ . What is its total surface area?
Medium
- A $200\pi cm^{3}$
- B $40\pi cm^{3}$
- C $80\pi cm^{3}$
- D $100\pi cm^{3}$

Summary

## 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.

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 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).

Slides

Volume of Cubes and Cuboids

Volume measures how much space a 3D shape takes up.
For a cuboid with length l, width w, height $h: V = lwh$ .
A cube is a special cuboid where $l = w = h$ .

A cuboid has length 8 cm, width 5 cm, height 11 cm. What is its volume?

$44 cm^{3}$
$400 cm^{3}$
$440 cm^{3}$
$880 cm^{3}$

Volume of Prisms

A prism has a constant cross-sectional area A and length l.
Volume $V = A \times l$ .
This formula works for any prism, including triangular prisms.

The volume of a prism is given by $V = ____ \times l$ , where A is the cross-sectional area.

Volume of Cylinders

A cylinder is like a prism with circular cross-section.
Volume $V = \pi r^{2}h$ , where r is radius and h is height.
This formula is given in the exam.

Find the volume of a cylinder with radius 5 cm and height 8 cm. Give your answer in terms of π.

$40\pi cm^{3}$
$80\pi cm^{3}$
$200\pi cm^{3}$
$400\pi cm^{3}$

Volume of Pyramids and Cones

Volume of a pyramid: $V = (\frac{1}{3}) \times$ base area × height.
Volume of a cone: $V = (\frac{1}{3})\pi r^{2}h$ .
Both formulas are given in the exam.

The volume of a cone is $V = (\frac{1}{3})\pi r^{2}____$ , where r is radius and h is perpendicular height.

Volume of Spheres

Volume of a sphere: $V = (\frac{4}{3})\pi r^{3}$ .
This formula is given in the exam.
A hemisphere has half the volume of a sphere.

What is the volume of a sphere with radius 6 cm? (Give answer in terms of π.)

$144\pi cm^{3}$
$216\pi cm^{3}$
$288\pi cm^{3}$
$864\pi cm^{3}$

Surface Area of Cuboids

Surface area is the sum of areas of all faces.
For a cuboid: $SA = 2(lw + lh + wh)$ .
For a cube with side $s: SA = 6s^{2}$ .

Which of these are true about surface area?

Surface area of a cuboid is $2(lw + lh + wh)$ .
Surface area of a cube with side 3 cm is $54 cm^{2}$ .
Surface area is measured in cubic units.

Surface Area of Cylinders

Curved surface area: 2πrh.
Total surface area: $2\pi rh + 2\pi r^{2}$ .
The curved surface area formula is given in the exam.

The total surface area of a cylinder is $2\pi rh + 2\pi r^{2}$ , where r is radius and h $is ____$ .

Surface Area of Cones and Spheres

Curved surface area of a cone: $\pi rl (l =$ slant height).
Total surface area of a cone: $\pi rl + \pi r^{2}$ .
Surface area of a sphere: $4\pi r^{2}$ .
These formulas are given in the exam.

A sphere has radius 5 cm. What is its surface area? (Give answer in terms of π.)

$25\pi cm^{2}$
$50\pi cm^{2}$
$100\pi cm^{2}$
$200\pi cm^{2}$

Problem Solving with Volumes

For compound shapes, find volumes of individual parts and add/subtract.
For a frustum, subtract the smaller cone/pyramid from the larger one.
Always check units and convert if necessary.

Which statements are correct?

A hemisphere is half a sphere.
Volume of a frustum = volume of larger cone + volume of smaller cone.
When solving volume problems, you may need to convert between units.

Examiner Tips

Only memorise the cuboid volume formula; others are given.
Read questions carefully: you may need to add extra areas (e.g., base).
Plan your approach before calculating.

A cylinder has radius 3 cm and height 4.5 cm. Calculate its total surface area in terms of π.

$27\pi cm^{2}$
$36\pi cm^{2}$
$45\pi cm^{2}$
$54\pi cm^{2}$

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