Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Calculate surface area of cubes and rectangular prisms.
• Calculate volume of cubes and rectangular prisms.
• Use appropriate units (cm², m², cm³, m³).
• Convert between different units of area and volume.
• Solve real-life word problems.

Lesson summary

This week, we delve deeper into the fascinating world of measurement, specifically focusing on calculating area, surface area, and volume. Measurement is crucial not just in mathematics class, but also in everyday life, from planning your backyard garden to understanding the size of a room for renovations or even calculating the amount of paint you need for a wall. In South Africa, with a vibrant construction industry and a rich history of architecture, understanding these concepts is even more relevant. We'll explore how these measurements help us understand the physical world around us and make informed decisions.

Lesson notes

2.1 Area Review Area is the amount of space a two-dimensional shape covers. It's measured in square units (e.g., cm², m²). We'll be using our understanding of area as the foundation for calculating surface area.

Remember the following formulas: Square: Area = side × side = s² Rectangle: Area = length × breadth = l × b 2.2 Surface Area Surface area is the total area of all the faces (or surfaces) of a three-dimensional object. Imagine unfolding a box – the surface area is the total area of the unfolded box. We will focus on cubes and rectangular prisms.

Cube: A cube has six identical square faces. If the side length of the cube is 's', then the area of one face is s².

Therefore, the surface area of a cube is 6 × s² = 6s².

Rectangular Prism: A rectangular prism has six rectangular faces. The opposite faces are identical. If the length is 'l', breadth is 'b', and height is 'h', then: Area of two faces is 2 × (l × b) Area of two other faces is 2 × (b × h) Area of the remaining two faces is 2 × (l × h) Therefore, the surface area of a rectangular prism is 2(lb + bh + lh).

Example 1: Surface Area of a Cube A wooden block used in a construction project is a cube with a side length of 15 cm. Calculate the surface area of the block.

Solution: Side length (s) = 15 cm Surface Area = 6s² = 6 × (15 cm)² = 6 × 225 cm² = 1350 cm² Therefore, the surface area of the cube is 1350 cm².

Example 2: Surface Area of a Rectangular Prism A cardboard box used for packaging measures 30 cm in length, 20 cm in breadth, and 10 cm in height. Calculate the amount of cardboard needed to make the box (i.e., the surface area).

Solution: Length (l) = 30 cm, Breadth (b) = 20 cm, Height (h) = 10 cm Surface Area = 2(lb + bh + lh) = 2((30 cm × 20 cm) + (20 cm × 10 cm) + (30 cm × 10 cm)) = 2(600 cm² + 200 cm² + 300 cm²) = 2(1100 cm²) = 2200 cm² Therefore, the surface area of the rectangular prism is 2200 cm². 2.3 Volume Volume is the amount of space a three-dimensional object occupies. It's measured in cubic units (e.g., cm³, m³).

Cube: Volume = side × side × side = s³ Rectangular Prism: Volume = length × breadth × height = l × b × h Example 3: Volume of a Cube Calculate the volume of a sugar cube with a side length of 1 cm.

Solution: Side length (s) = 1 cm Volume = s³ = (1 cm)³ = 1 cm³ Therefore, the volume of the cube is 1 cm³.

Example 4: Volume of a Rectangular Prism A water tank is in the shape of a rectangular prism with a length of 2 m, a breadth of 1.5 m, and a height of 1 m. Calculate the volume of water the tank can hold.

Solution: Length (l) = 2 m, Breadth (b) = 1.5 m, Height (h) = 1 m Volume = l × b × h = 2 m × 1.5 m × 1 m = 3 m³ Therefore, the volume of the water tank is 3 m³. 2.4 Units and Conversions It's crucial to use consistent units when calculating area, surface area, and volume. Here are some common conversions within the metric system: Length: 1 m = 100 cm, 1 cm = 10 mm Area: 1 m² = (100 cm)² = 10,000 cm², 1 cm² = (10 mm)² = 100 mm² Volume: 1 m³ = (100 cm)³ = 1,000,000 cm³, 1 cm³ = (10 mm)³ = 1,000 mm³ Example 5: Unit Conversion Convert 5 m² to cm².

Solution: We know that 1 m² = 10,000 cm².

Therefore, 5 m² = 5 × 10,000 cm² = 50,000 cm². Guided Practice (With Solutions)

Question 1: A cube has a side length of 8 cm. Find its surface area.

Solution: Surface Area = 6s² = 6 × (8 cm)² = 6 × 64 cm² = 384 cm²

Commentary:* We use the formula for the surface area of a cube (6s²) and substitute the given side length. Remember to include the units (cm²).

Question 2: A rectangular prism has a length of 5 cm, a breadth of 4 cm, and a height of 3 cm. Find its volume.

Solution: Volume = l × b × h = 5 cm × 4 cm × 3 cm = 60 cm³

Commentary:* This is a direct application of the formula for the volume of a rectangular prism. The units are cubic centimeters (cm³).

Question 3: A rectangular prism has a length of 2 m, a breadth of 1.5 m, and a height of 0.5 m. Find its surface area.

Solution: Surface Area = 2(lb + bh + lh) = 2((2 m × 1.5 m) + (1.5 m × 0.5 m) + (2 m × 0.5 m)) = 2(3 m² + 0.75 m² + 1 m²) = 2(4.75 m²) = 9.5 m²

Commentary:* Again, use the formula carefully, ensuring you multiply and add the areas of the faces correctly.

Question 4: Convert 2 m³ to cm³.

Solution: 1 m³ = 1,000,000 cm³ 2 m³ = 2 × 1,000,000 cm³ = 2,000,000 cm³

Commentary:* Remember the conversion factor between m³ and cm³. Independent Practice (Questions Only) Calculate the surface area of a cube with a side length of 6 cm. Calculate the volume of a rectangular prism with dimensions 10 cm x 5 cm x 2 cm. A rectangular prism has a length of 4 m, a breadth of 3 m, and a height of 2 m. Find its surface area. A container in the shape of a cube has sides of 0.5m. What is the volume of the container in m³? Convert 3 m³ to cm³. A room is 5 m long, 4 m wide, and 3 m high. What is the volume of the room? A rectangular prism has a length of 8 cm, a breadth of 5 cm, and a height of 2 cm. Find its surface area. A water tank is in the shape of a cube with side 1.2m.