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Exploring Perimeter, Area, and Volume in Math Education

Discover the fundamentals of calculating perimeter, area, volume, and surface area in geometry. Dive into concepts such as calculating cubic units, using formulas, and solving practical math problems.

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Exploring Perimeter, Area, and Volume in Math Education

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  1. S8 Perimeter, area and volume Contents S8.1 Perimeter • A S8.2 Area • A S8.3 Surface area • A S8.4 Volume • A S8.5 Circumference of a circle • A S8.6 Area of a circle • A

  2. Making cuboids The following cuboid is made out of interlocking cubes. How many cubes does it contain?

  3. Making cuboids We can work this out by dividing the cuboid into layers. The number of cubes in each layer can be found by multiplying the number of cubes along the length by the number of cubes along the width. 3 × 4 = 12 cubes in each layer. There are three layers altogether so the total number of cubes in the cuboid = 3 × 12 = 36 cubes.

  4. Making cuboids The amount of space that a three-dimensional object takes up is called its volume. Volume is measured in cubic units. We can use mm3, cm3, m3 or km3. The 3 tells us that there are three dimensions, length, width and height. Liquid volume or capacity is measured in ml, l, pints or gallons.

  5. Volume of a cuboid Volume of a cuboid = length × width × height = lwh We can find the volume of a cuboid by multiplying the area of the base by the height. The area of the base = length × width So: height, h length, l width, w

  6. Volume of a cuboid What is the volume of this cuboid? Volume of cuboid = length × width × height 5 cm = 13 × 8 × 5 13 cm 8 cm = 520 cm3

  7. Volume of a prism made from cuboids What is the volume of this L-shaped prism? 3 cm We can think of the shape as two cuboids joined together. 3 cm Volume of the green cuboid 4 cm = 6 × 3 × 3 = 54 cm3 6 cm Volume of the blue cuboid = 3 × 2 × 2 = 12 cm3 Total volume 5 cm = 54 + 12 = 66 cm3

  8. Volume of a prism Volume of a prism = area of cross-section × length Remember, a prism is a 3-D shape with the same cross-section throughout its length. 3 cm We can think of this prism as lots of L-shaped surfaces running along the length of the shape. If the cross-section has an area of 22 cm2 and the length is 3 cm: Volume of L-shaped prism = 22 × 3 = 66 cm3

  9. Volume of a prism What is the volume of this triangular prism? 7.2 cm 4 cm 5 cm Area of cross-section = ½ × 5 × 4 = 10 cm2 Volume of prism = 10 × 7.2 = 72 cm3

  10. Volume of a prism What is the volume of this prism? 12 m 4 m 7 m 3 m 5 m Area of cross-section = 7 × 12 – 4 × 3 = 84 – 12 = 72 m2 Volume of prism = 5 × 72 = 360 m3

  11. S8 Perimeter, area and volume Contents S8.1 Perimeter • A S8.2 Area • A • A S8.3 Surface area S8.4 Volume • A S8.5 Circumference of a circle • A S8.6 Area of a circle • A

  12. Surface area of a cuboid To find the surfacearea of a shape, we calculate the total area of all of the faces. A cuboid has 6 faces. The top and the bottom of the cuboid have the same area.

  13. Surface area of a cuboid To find the surfacearea of a shape, we calculate the total area of all of the faces. A cuboid has 6 faces. The front and the back of the cuboid have the same area.

  14. Surface area of a cuboid To find the surfacearea of a shape, we calculate the total area of all of the faces. A cuboid has 6 faces. The left hand side and the right hand side of the cuboid have the same area.

  15. Surface area of a cuboid To find the surfacearea of a shape, we calculate the total area of all of the faces. Can you work out the surface area of this cuboid? 5 cm 8 cm The area of the top = 8 × 5 = 40 cm2 7 cm The area of the front = 7 × 5 = 35 cm2 The area of the side = 7 × 8 = 56 cm2

  16. Surface area of a cuboid To find the surfacearea of a shape, we calculate the total area of all of the faces. So the total surface area = 5 cm 8 cm 2 × 40 cm2 Top and bottom 7 cm + 2 × 35 cm2 Front and back + 2 × 56 cm2 Left and right side = 80 + 70 + 112 = 262 cm2

  17. Formula for the surface area of a cuboid w l 2 × lw Top and bottom Front and back + 2 × hw h + 2 × lh Left and right side We can find the formula for the surface area of a cuboid as follows. Surface area of a cuboid = = 2lw + 2hw + 2lh

  18. Surface area of a cube x Surface area of a cube = 6x2 How can we find the surface area of a cube of length x? All six faces of a cube have the same area. The area of each face is x × x = x2. Therefore:

  19. Chequered cuboid problem This cuboid is made from alternate purple and green centimetre cubes. What is its surface area? Surface area = 2 × 3 × 4 + 2 × 3 × 5 + 2 × 4 × 5 = 24 + 30 + 40 = 94 cm2 How much of the surface area is green? 47 cm2

  20. Surface area of a prism What is the surface area of this L-shaped prism? 3 cm To find the surface area of this shape we need to add together the area of the two L-shapes and the area of the six rectangles that make up the surface of the shape. 3 cm 4 cm 6 cm Total surface area = 2 × 22 + 18 + 9 + 12 + 6 + 6 + 15 5 cm = 110 cm2

  21. Using nets to find surface area 6 cm 3 cm 3 cm 6 cm 5 cm 3 cm 3 cm It can be helpful to use the net of a 3-D shape to calculate its surface area. Here is the net of a 3 cm by 5 cm by 6 cm cuboid. Write down the area of each face. 18 cm2 Then add the areas together to find the surface area. 15 cm2 15 cm2 30 cm2 30 cm2 18 cm2 Surface Area = 126 cm2

  22. Using nets to find surface area Here is the net of a regular tetrahedron. What is its surface area? Area of each face = ½bh = ½ × 6 × 5.2 = 15.6 cm2 5.2 cm Surface area = 4 × 15.6 = 62.4 cm2 6 cm

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