Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Calculate surface area and volume of prisms, cylinders, pyramids, and cones.
• Convert between different units of measurement.
• Apply these concepts to solve real-world problems.
• Understand the effect of scaling on surface area and volume.

Lesson summary

Measurement is a fundamental skill in mathematics and in everyday life. From calculating the amount of paint needed for a room to understanding the volume of water in a dam, measurement plays a crucial role. In South Africa, where resources are often limited and careful planning is essential, accurate measurement is even more important. Farmers need to measure land area for planting crops, builders need to calculate the amount of material needed for construction projects, and cooks need to measure ingredients for recipes. This week, we will focus on calculating the surface area and volume of various 3D objects, skills that are vital in numerous professional and personal contexts.

Lesson notes

2.1 Surface Area Surface area is the total area of all the surfaces of a 3D object. It is measured in square units (e.g., cm², m², km²). Imagine painting the entire outside of an object; the surface area is the amount of paint you would need to cover it completely.

Prism (Cube and Cuboid): A prism is a 3D shape with two identical ends (bases) and flat rectangular sides.

Cube: A cube has six identical square faces. If the side length of a cube is s, then its surface area is 6s².

Cuboid: A cuboid has six rectangular faces. If the length is l, width is w, and height is h, then its surface area is 2(lw + lh + wh).

Cylinder: A cylinder has two circular bases and a curved surface. If the radius of the base is r and the height is h, then its surface area is 2πr² + 2πrh. (2πr² accounts for both circular ends and 2πrh accounts for the curved surface.)

Pyramid: A pyramid has a polygonal base and triangular faces that meet at a point (apex). The surface area depends on the shape of the base and the slant height of the triangular faces. For a square-based pyramid with base side a and slant height l, the surface area is a² + 2al. (a² accounts for the base and 2a*l accounts for the 4 triangle faces).

Cone: A cone has a circular base and a curved surface that tapers to a point. If the radius of the base is r and the slant height is l, then its surface area is πr² + πrl. (πr² accounts for the base and πrl accounts for the curved surface.) 2.2 Volume Volume is the amount of space a 3D object occupies. It is measured in cubic units (e.g., cm³, m³, km³). Imagine filling an object with water; the volume is the amount of water it can hold.

Prism (Cube and Cuboid): Cube: If the side length of a cube is s, then its volume is s³.

Cuboid: If the length is l, width is w, and height is h, then its volume is lwh.

Cylinder: If the radius of the base is r and the height is h, then its volume is πr²h.

Pyramid: The volume of a pyramid is (1/3) (Base Area) Height. For a square-based pyramid with base side a and perpendicular height h, the volume is (1/3)a²*h.

Cone: The volume of a cone is (1/3)πr²h, where r is the radius of the base and h is the perpendicular height. 2.3 Units of Measurement and Conversions It's crucial to be comfortable converting between different units.

Here are some common conversions: Length: 1 cm = 10 mm 1 m = 100 cm 1 km = 1000 m Volume/Capacity: 1 litre (l) = 1000 millilitres (ml) 1 kilolitre (kl) = 1000 litres (l) 1 cm³ = 1 ml 1 m³ = 1000 litres Mass: 1 kg = 1000 g 1 tonne = 1000 kg 2.4 Scaling When the dimensions of a 3D object are scaled (multiplied by a factor), the surface area changes by the square of the scaling factor, and the volume changes by the cube of the scaling factor.

Example: If you double the side length of a cube, the surface area becomes 4 times larger (2² = 4) and the volume becomes 8 times larger (2³ = 8).

Worked example

Example 1: Surface Area of a Cylinder

A cylindrical water tank has a radius of 1.5 m and a height of 4 m. Calculate the surface area of the tank.

Solution:

Surface Area = 2πr² + 2πrh

Surface Area = 2 π (1.5 m)² + 2 π (1.5 m) * (4 m)

Surface Area = 2 π 2.25 m² + 2 π 6 m²

Surface Area = 4.5π m² + 12π m²

Surface Area = 16.5π m²

Surface Area ≈ 51.84 m²