Lesson Notes By Weeks and Term v5 - Grade 8
Measurement: area, surface area and volume (Grade 8) – Week 2 focus
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Measurement: area, surface area and volume (Grade 8) – Week 2 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematics
Class: Grade 8
Term: 3rd Term
Week: 2
Theme: General lesson support
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This week, we delve deeper into the world of measurement, specifically focusing on area, surface area, and volume. These concepts are fundamental not only in mathematics but also in everyday life. Imagine planning a garden, building a shack, or even just figuring out how much paint you need for your room - all require an understanding of area, surface area, and volume. These skills are also crucial for many careers, from construction and engineering to architecture and design. In South Africa, with its diverse landscapes and growing infrastructure, understanding measurement is particularly important.
2.1 Area of Compound Shapes A compound shape is a shape made up of two or more simpler shapes joined together. To find the area of a compound shape, we break it down into these simpler shapes (usually rectangles, squares, and triangles), calculate the area of each individual shape, and then add the areas together.
Example 1: Consider the shape below, with dimensions in centimeters: ``` _______ | | 4 cm |_______| / \ / \ 3 cm /____________\ 8 cm ``` This shape is a rectangle sitting on top of a triangle.
Rectangle Area: Length x Width = 8 cm x 4 cm = 32 cm² Triangle Area: (1/2) x Base x Height = (1/2) x 8 cm x 3 cm = 12 cm² Total Area: 32 cm² + 12 cm² = 44 cm² Why does this work? We are essentially dividing the complex shape into manageable parts, finding the area of each part, and then summing them to get the total area. 2.2 Surface Area of Cubes and Rectangular Prisms Surface area is the total area of all the faces of a 3D object.
Cube: A cube has 6 identical square faces. If the side length of the cube is 's', then the area of each face is s², and the surface area of the cube is 6s².
Rectangular Prism (Cuboid): A rectangular prism has 6 rectangular faces. If the length is 'l', the width is 'w', and the height is 'h', then the surface area is 2(lw + lh + wh). This is because there are two faces with area lw, two with area lh, and two with area wh.
Example 2 (Cube): A cube has a side length of 5 cm. Calculate its surface area.
Area of one face: 5 cm x 5 cm = 25 cm² Surface area: 6 x 25 cm² = 150 cm² Example 3 (Rectangular Prism): A rectangular prism has a length of 7 cm, a width of 3 cm, and a height of 4 cm. Calculate its surface area. Surface area = 2(lw + lh + wh) = 2((7 cm x 3 cm) + (7 cm x 4 cm) + (3 cm x 4 cm)) Surface area = 2(21 cm² + 28 cm² + 12 cm²) = 2(61 cm²) = 122 cm² Why is this important? Understanding surface area helps us calculate how much material we need to cover a 3D object, like wrapping a gift or painting a room. 2.3 Volume of Cubes and Rectangular Prisms Volume is the amount of space a 3D object occupies.
Cube: If the side length of the cube is 's', then the volume is s³.
Rectangular Prism: If the length is 'l', the width is 'w', and the height is 'h', then the volume is lwh.
Example 4 (Cube): A cube has a side length of 6 cm. Calculate its volume. Volume = 6 cm x 6 cm x 6 cm = 216 cm³ Example 5 (Rectangular Prism): A rectangular prism has a length of 8 cm, a width of 4 cm, and a height of 2 cm. Calculate its volume. Volume = 8 cm x 4 cm x 2 cm = 64 cm³ Why is this important? Volume helps us determine how much we can fit inside a container, like the capacity of a water tank or the amount of sand in a bucket. 2.4 Unit Conversions It is crucial to be able to convert between different units of measurement.
Area: 1 m = 100 cm, so 1 m² = (100 cm)² = 10,000 cm² Volume: 1 m = 100 cm, so 1 m³ = (100 cm)³ = 1,000,000 cm³ Example 6: Convert 5 m² to cm². 5 m² = 5 x 10,000 cm² = 50,000 cm² Example 7: Convert 2,000,000 cm³ to m³. 2,000,000 cm³ = 2,000,000 / 1,000,000 m³ = 2 m³ Why convert units? Often, measurements are given in different units, and we need to convert them to a common unit before performing calculations or comparisons. Guided Practice (With Solutions)
Question 1: A compound shape is formed by a square with sides of 6 cm attached to a semi-circle with a diameter of 6 cm along one of the square's sides. Calculate the total area of the compound shape.
Solution: Square Area: side x side = 6 cm x 6 cm = 36 cm² Semi-circle Area: (1/2) π r² = (1/2) π (3 cm)² ≈ (1/2) 3.14159 * 9 cm² ≈ 14.14 cm² Total Area: 36 cm² + 14.14 cm² ≈ 50.14 cm²
Commentary: We broke the compound shape into a square and a semi-circle. We used the appropriate formulas for the area of each and then added them together. We use π ≈ 3.
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9. Question 2: A rectangular prism has a length of 10 cm, a width of 5 cm, and a height of 3 cm. Calculate its surface area and volume.
Solution: Surface Area: 2(lw + lh + wh) = 2((10 cm x 5 cm) + (10 cm x 3 cm) + (5 cm x 3 cm)) = 2(50 cm² + 30 cm² + 15 cm²) = 2(95 cm²) = 190 cm² Volume: lwh = 10 cm x 5 cm x 3 cm = 150 cm³
Commentary: We used the formulas for surface area and volume of a rectangular prism directly, substituting the given values.
Question 3: Convert 0.5 m³ to cm³.
Solution: 1 m³ = 1,000,000 cm³ 0.5 m³ = 0.5 x 1,000,000 cm³ = 500,000 cm³
Commentary: We multiplied the given volume in m³ by the conversion factor to obtain the volume in cm³. Independent Practice (Questions Only) A compound shape is formed by a rectangle (12 cm x 5 cm) and a triangle (base = 5 cm, height = 4 cm). Calculate the total area. Calculate the surface area of a cube with sides of 8 cm. Calculate the volume of a rectangular prism with length = 15 cm, width = 6 cm, and height = 4 cm. A swimming pool is 5m long, 3m wide, and 1.5m deep. How many cubic meters of water can it hold? Convert 3 m³ to cm³. Convert 750,000 cm³ to m³. A rectangular block of wood has dimensions 20cm x 10cm x 5cm.