Surface Area and Volume Cubes and Cuboids

Surface area of a cuboid To find the surface area 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. Discuss the meaning of surface area. The important thing to remember is that although surface area is found for three-dimensional shapes, surface area only has two dimensions. It is therefore measured in square units.

Surface area of a cuboid To find the surface area 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.

Surface area of a cuboid To find the surface area 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.

Surface area of a cuboid To find the surface area of a shape, we calculate the total area of all of the faces. Can you work out the surface area of this cubiod? 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

Surface area of a cuboid To find the surface area 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 Stress the importance to work systematically when finding the surface area to ensure that no faces have been left out. We can also work out the surface area of a cuboid by drawing its net (see slide 51). This may be easier for some pupils because they would be able to see every face rather than visualizing it. + 2 × 56 cm2 Left and right side = 80 + 70 + 112 = 262 cm2

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

How can we find the surface area of a cube of length x? All six faces of a cube have the same area. x The area of each face is x × x = x2 Therefore, As pupils to use this formula to find the surface area of a cube of side length 5 cm. 6 × 52 = 6 × 25 = 150 cm2. Repeat for other numbers. As a more challenging question tell pupils that a cube has a surface area of 96 cm2. Ask them how we could work out its side length using inverse operations. Surface area of a cube = 6x2

Checkered 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 Discuss how to work out the surface area that is green. Ask pupils how we could write the proportion of the surface area that is green as a fraction, as a decimal and as a percentage. How much of the surface area is green? 48 cm2

What is the surface area of this L-shaped prism? 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 6 rectangles that make up the surface of the shape. 3 cm 4 cm 6 cm Discuss ways to find the surface area of this solid. We could use a net of this prism to help find the area of each face. Total surface area = 2 × 22 + 18 + 9 + 12 + 6 + 6 + 15 5 cm = 110 cm2

Using nets to find surface area 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 cubiod. 5 cm 6 cm 3 cm Write down the area of each face. 18 cm2 Then add the areas together to find the surface area. 15 cm2 Links: S3 3-D shapes – nets S6 Construction and Loci – constructing nets 30 cm2 15 cm2 30 cm2 18 cm2 Surface Area = 126 cm2

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

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

Making cuboids The amount of space that a three-dimensional object takes up is called its volume. Volume is measured in cubic units. For example, we can use mm3, cm3, m3 or km3. The 3 tells us that there are three dimensions, length, width and height. Link: S7 Measures – units of volume and capacity Liquid volume or capacity is measured in ml, l, pints or gallons.

Volume of a cuboid 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 Volume of a cuboid = length × width × height = lwh length, l width, w

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

Volume and displacement Ask pupils how we could use water in a measuring cylinder to find the volume of an object. Tell pupils that 1 cm3 of water will displace 1 ml of water in the beaker. Demonstrate this by dropping each cuboid into the beaker, and recording how the level of the water changes. Use this slide to demonstrate how volume is linked to capacity. Links: S7 Measures – units of volume and capacity S7 Measures – reading scales

Volume and displacement By dropping cubes and cuboids into a measuring cylinder half filled with water we can see the connection between the volume of the shape and the volume of the water displaced. 1 ml of water has a volume of 1 cm3 For example, if an object is dropped into a measuring cylinder and displaces 5 ml of water then the volume of the object is 5 cm3. Ask pupils to give the dimensions of a cube that would hold 1 litre of water. This would be a 10 cm by 10 cm by 10 cm cube. Ask pupils how many litres of water we could fit into a metre cube. (1000 litres). A litre of water has a weight of 1 kg. A metre cube would therefore hold 1 tonne of water! Link: S7 Measures – units of volume and capacity What is the volume of 1 litre of water? 1 litre of water has a volume of 1000 cm3.

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 Compare this with slide 50, which finds the surface area of the same shape. = 3 × 2 × 2 = 12 cm3 Total volume 5 cm = 54 + 12 = 66 cm3

Volume of a prism 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. Volume of a prism = area of cross-section × length 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

What is the volume of this prism? 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