The aim of this unit is to teach pupils to: Identify and use the geometric properties of triangles, quadrilaterals and other polygons to solve problems; explain and justify inferences and deductions using mathematical reasoning Understand congruence and similarity Identify and use the properties of circles Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp 184-197.
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.
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.
Pupils should write this formula down.
As pupils to use this formula to find the surface area of a cube of side length 5 cm. 6 × 5 2 = 6 × 25 = 150 cm 2 . Repeat for other numbers. As a more challenging question tell pupils that a cube has a surface area of 96 cm 2 . Ask them how we could work out its side length using inverse operations.
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.
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.
Links: S3 3-D shapes – nets S6 Construction and Loci – constructing nets
Link: S7 Measures – units of volume and capacity
Ask pupils how we could use water in a measuring cylinder to find the volume of an object. Tell pupils that 1 cm 3 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
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
Compare this with slide 50, which finds the surface area of the same shape.
Surface area and volume of cuboids
Shape and Space Cuboids
Surface area of a cuboidTo find the surface area of a shape, we calculate thetotal area of all of the faces. A cuboid has 6 faces. The top and the bottom of the cuboid have the same area.
Surface area of a cuboidTo find the surface area of a shape, we calculate thetotal 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 cuboidTo find the surface area of a shape, we calculate thetotal 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 5 cm 8 cm surface area of this cuboid? The area of the top = 8 × 5 = 40 cm27 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. 5 cm So the total surface area = 8 cm 2 × 40 cm2 Top and bottom7 cm + 2 × 35 cm2 Front and back + 2 × 56 cm2 Left and right side = 80 + 70 + 112 = 262 cm2
Formula for the surface area of a cuboidWe can find the formula for the surface area of a cuboidas follows. Surface area of a cuboid = w l 2 × lw Top and bottom h + 2 × hw Front and back + 2 × lh Left and right side = 2lw + 2hw + 2lh
Surface area of a cube 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, x Surface area of a cube = 6x2
Chequered cuboid problemThis cuboid is made from alternate purple and greencentimetre 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? 48 cm2
Surface area of a prism What is the surface area of this L-shaped prism? 3 cm To find the surface area of 3 cm this shape we need to add together the area of the two 4 cm L-shapes and the area of the 6 rectangles that make up6 cm the surface of the shape. 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 cuboid 6 cm Write down the area of each 3 cm 18 cm2 3 cm 6 cm face. Then add the5 cm 15 cm2 30 cm2 15 cm2 30 cm2 areas together to find the surface area. 3 cm 18 cm2 3 cm Surface Area = 126 cm2
Making cuboidsThe following cuboid is made out of interlocking cubes. How many cubes does it contain?
Making cuboidsWe 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 cuboidsThe amount of space that a three-dimensional object takesup 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, widthand height.Liquid volume or capacity is measured in ml, l, pints orgallons.
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 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 cm 3 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. What is the volume of 1 litre of water? 1 litre of water has a volume of 1000 cm 3.
Volume of a prism made from cuboids What is the volume of this L-shaped prism? 3 cm 3 cm We can think of the shape as two cuboids joined together. 4 cm Volume of the green cuboid = 6 × 3 × 3 = 54 cm36 cm Volume of the blue cuboid = 3 × 2 × 2 = 12 cm3 Total volume 5 cm = 54 + 12 = 66 cm3
Volume of a prismRemember, a prism is a 3-D shape with the samecross-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
Volume of a prism What is the volume of this prism? 12 m 4m 7m 3m 5m Area of cross-section = 7 × 12 – 4 × 3 = 84 – 12 = 72 m2 Volume of prism = 5 × 72 = 360 m3