Geometry & measures

Area, Surface Area and Volume of 2D and 3D Shapes

Year 10 · Year 11

✓By the end of this lesson students will be able to calculate the circumference and area of circles.
✓By the end of this lesson students will be able to calculate the arc length and area of sectors.
✓By the end of this lesson students will be able to calculate the volume and surface area of prisms and cylinders.
✓By the end of this lesson students will be able to calculate the volume and surface area of cones and spheres.
✓By the end of this lesson students will be able to solve problems involving the area, surface area, and volume of composite 2D and 3D shapes.

Key concepts

Circle

A 2D shape where all points on the circumference are equidistant from the centre. The radius (r) is the distance from the centre to the circumference, and the diameter (d) is the distance across the circle through the centre (d = 2r). Pi (π) is a mathematical constant, approximately 3.14159.

Circumference (C) = πd = 2πr, Area (A) = πr²

Sector

A part of a circle enclosed by two radii and an arc. The angle (θ) is the angle at the centre of the circle, measured in degrees.

Arc Length (L) = (θ/360) × 2πr, Area (A) = (θ/360) × πr²

Prism

A 3D shape with the same cross-section throughout its length. Examples include cuboids, triangular prisms, and cylinders (a circular prism).

Volume (V) = Area of cross-section × length, Surface Area (SA) = 2 × Area of cross-section + Perimeter of cross-section × length

Cylinder

A 3D shape with a circular cross-section, which is a specific type of prism. 'h' represents the height of the cylinder.

Volume (V) = πr²h, Total Surface Area (SA) = 2πr² + 2πrh (area of two circular ends + curved surface area)

Cone

A 3D shape with a circular base and a single vertex. 'h' is the perpendicular height from the base to the vertex, and 'l' is the slant height along the curved surface. The relationship between r, h, and l is given by Pythagoras' theorem: l² = r² + h².

Volume (V) = (1/3)πr²h, Curved Surface Area (CSA) = πrl, Total Surface Area (TSA) = πrl + πr²

Sphere

A perfectly round 3D object where every point on its surface is equidistant from its centre. 'r' is the radius of the sphere.

Volume (V) = (4/3)πr³, Surface Area (SA) = 4πr²

Key facts to remember

1Circumference of a circle: C = 2πr or C = πd
2Area of a circle: A = πr²
3Arc length of a sector: L = (θ/360) × 2πr
4Area of a sector: A = (θ/360) × πr²
5Volume of a prism: V = Area of cross-section × length
6Volume of a cylinder: V = πr²h
7Total surface area of a cylinder: SA = 2πr² + 2πrh
8Volume of a cone: V = (1/3)πr²h
9Curved surface area of a cone: CSA = πrl (where l is slant height)
10Volume of a sphere: V = (4/3)πr³
11Surface area of a sphere: SA = 4πr²
12Pythagoras' theorem for cones: l² = r² + h²

Worked examples

Example 1

A circle has a radius of 7 cm. Calculate its circumference and area. A sector of this circle has an angle of 120° at the centre. Calculate its arc length and area.

I1. Calculate the circumference of the circle:

II C = 2πr = 2 × π × 7 = 14π cm

III C ≈ 43.98 cm (to 2 decimal places)

IV2. Calculate the area of the circle:

V A = πr² = π × 7² = 49π cm²

VI A ≈ 153.94 cm² (to 2 decimal places)

VII3. Calculate the arc length of the sector:

VIII L = (θ/360) × 2πr = (120/360) × 2 × π × 7

9 L = (1/3) × 14π = (14/3)π cm

10 L ≈ 14.66 cm (to 2 decimal places)

114. Calculate the area of the sector:

12 A_sector = (θ/360) × πr² = (120/360) × π × 7²

13 A_sector = (1/3) × 49π = (49/3)π cm²

14 A_sector ≈ 51.31 cm² (to 2 decimal places)

Answer

Circumference = 14π cm (or 43.98 cm), Area = 49π cm² (or 153.94 cm²), Arc Length = (14/3)π cm (or 14.66 cm), Sector Area = (49/3)π cm² (or 51.31 cm²).

It is often best to leave answers in terms of π until the final step for maximum accuracy, then round as required.

Example 2

A cylindrical water tank has a radius of 1.5 m and a height of 3 m. Calculate its volume and total surface area.

I1. Calculate the volume of the cylinder:

II V = πr²h = π × (1.5)² × 3

III V = π × 2.25 × 3 = 6.75π m³

IV V ≈ 21.21 m³ (to 2 decimal places)

V2. Calculate the total surface area of the cylinder:

VI SA = 2πr² + 2πrh

VII SA = 2π(1.5)² + 2π(1.5)(3)

VIII SA = 2π(2.25) + 2π(4.5)

9 SA = 4.5π + 9π = 13.5π m²

10 SA ≈ 42.41 m² (to 2 decimal places)

Answer

Volume = 6.75π m³ (or 21.21 m³), Total Surface Area = 13.5π m² (or 42.41 m²).

Remember the total surface area includes the two circular ends and the curved surface.

Example 3

A solid cone has a radius of 5 cm and a perpendicular height of 12 cm. Calculate its volume and total surface area. A sphere has a radius of 6 cm. Calculate its volume and surface area.

I1. For the cone, first find the slant height (l) using Pythagoras' theorem:

II l² = r² + h² = 5² + 12² = 25 + 144 = 169

III l = √169 = 13 cm

IV2. Calculate the volume of the cone:

V V_cone = (1/3)πr²h = (1/3) × π × 5² × 12

VI V_cone = (1/3) × π × 25 × 12 = 100π cm³

VII V_cone ≈ 314.16 cm³ (to 2 decimal places)

VIII3. Calculate the total surface area of the cone:

9 TSA_cone = πrl + πr² = π(5)(13) + π(5)²

10 TSA_cone = 65π + 25π = 90π cm²

11 TSA_cone ≈ 282.74 cm² (to 2 decimal places)

124. For the sphere, calculate its volume:

13 V_sphere = (4/3)πr³ = (4/3) × π × 6³

14 V_sphere = (4/3) × π × 216 = 4 × 72π = 288π cm³

15 V_sphere ≈ 904.78 cm³ (to 2 decimal places)

165. Calculate the surface area of the sphere:

17 SA_sphere = 4πr² = 4 × π × 6²

18 SA_sphere = 4 × π × 36 = 144π cm²

19 SA_sphere ≈ 452.39 cm² (to 2 decimal places)

Answer

Cone Volume = 100π cm³ (or 314.16 cm³), Cone Total Surface Area = 90π cm² (or 282.74 cm²). Sphere Volume = 288π cm³ (or 904.78 cm³), Sphere Surface Area = 144π cm² (or 452.39 cm²).

Always identify if perpendicular height or slant height is given for cones, and use Pythagoras' theorem to find the missing length if needed.

Common mistakes

✗Confusing radius (r) and diameter (d). Remember d = 2r.
✗Using the wrong formula for area, circumference, volume, or surface area.
✗Forgetting to include all faces when calculating total surface area (e.g., the two circular ends of a cylinder, or the base of a cone).
✗Rounding intermediate steps too early, which can lead to inaccuracies in the final answer. Keep answers in terms of π until the final step, or use the full calculator value.
✗Incorrectly using units (e.g., cm instead of cm² for area, or cm³ for volume).

Exam tips

★Always write down the formula you are using before substituting values. This shows your method and can earn marks even if you make a calculation error.
★Use the π button on your calculator for accuracy, rather than an approximation like 3.14.
★Pay close attention to the units given in the question and ensure your final answer has the correct units (e.g., cm, cm², cm³).
★Read the question carefully to determine if you need to calculate circumference, area, volume, or surface area, and whether it's a full shape or a part (like a sector or hemisphere).
★For questions involving slant height (l) in cones, remember to use Pythagoras' theorem (l² = r² + h²) if l is not given directly.

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