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**S8 Perimeter, area and volume**

KS3 Mathematics The aim of this unit is to teach pupils to: Deduce and use formulae to calculate lengths, perimeters, areas and volumes in 2-D and 3-D shapes Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp S8 Perimeter, area and volume

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**S8 Perimeter, area and volume**

Contents S8 Perimeter, area and volume A 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

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**Put these shapes in order**

Ask pupils to explain what perimeter is and how we could work out the perimeters of the shapes shown on the board. Challenge pupils to put these shapes in order from the one that has the smallest perimeter to the one that has the largest. Establish that it is not necessary to know the exact length of the diagonal edge. Ask pupils how we know that the diagonal edges must be between 1 and 2 units long. When the activity has been completed ask, If we put these shapes in order of area from smallest to largest, would the order be the same? Establish that the order would be different because the area of a shape is not directly related to its perimeter. Ask pupils to put the shapes in order of area.

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**What is the perimeter of this shape?**

To find the perimeter of a shape we add together the length of all the sides. What is the perimeter of this shape? Starting point 1 cm 3 Perimeter = = 12 cm 2 3 Ask pupils if they know how many dimensions measurements of perimeter have. Establish that they only have one dimension, length, even though the measurement is used for two-dimensional shapes. Tell pupils that when finding the perimeter of a shape with many sides it is a good idea to mark a starting point and then work from there adding up the lengths of all the sides. 1 1 2

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**Perimeter of a rectangle**

To calculate the perimeter of a rectangle we can use a formula. length, l width, w Using l for length and w for width, Explain the difference between the two forms of the formula. The first formula means double the length, double the width and add the two together. The second formula means add the length and the width and double the answer. Perimeter of a rectangle = l + w + l + w = 2l + 2w or = 2(l + w)

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**What is the perimeter of this shape?**

Sometimes we are not given the lengths of all the sides. We have to work them out using the information we are given. For example, 9 cm 5 cm 12 cm 4 cm What is the perimeter of this shape? The lengths of two of the sides are not given so we have to work them out before we can find the perimeter. Stress that to work out the perimeter we need to add together the lengths of every side. If we are not given some of the lengths, then we have to work them out before we can find the perimeter. a cm Let’s call the lengths a and b. b cm

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Perimeter Sometime we are not given the lengths of all the sides. We have to work them out from the information we are given. 9 cm 12 – 5 cm a = = 7 cm 5 cm b = 9 – 4 cm = 5 cm 12 cm 4 cm Discuss how to work out the missing sides of this shape. The side marked a cm plus the 5 cm side must be equal to 12 cm, a is therefore 7 cm. The side marked b cm plus the 4 cm side must be equal to 9 cm, b is therefore 5 cm. Can anyone in the class see a quicker way of working out the perimeter of this shape? a cm 7 cm P = = 42 cm 5 cm b cm

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**Calculate the lengths of the missing sides to find the perimeter.**

5 cm p = 2 cm p 2 cm q = r = 1.5 cm q r s = 6 cm t = 2 cm s 6 cm u = 10 cm Discuss how to find each missing length. 4 cm 4 cm P = 2 cm t 2 cm u = 46 cm

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**What is the perimeter of this shape?**

Remember, the dashes indicate the sides that are the same length. 5 cm 4 cm P = = 26 cm Rather than using repeated addition, pupils might suggest 2 × × 4 = 26 cm.

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**What is the perimeter of this shape?**

Start by finding the lengths of all the sides. 4.5 m 4.5 cm Perimeter = 5 m 4 m = 17 cm Discuss how pupils have calculated the lengths of the missing sides. 2 m 2 m 1 cm 1 cm 2 m

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**What is the perimeter of this shape?**

Before we can find the perimeter we must convert all the lengths to the same units. 256 cm In this example, we can either use metres or centimetres. 300 cm 3 m 1.9 m 190 cm Using centimetres, 240 cm 2.4 m P = = 986 cm

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**Which shape has a different perimeter from the first shape?**

Equal perimeters Which shape has a different perimeter from the first shape? A B C B A B C A Ask pupils to decide which shape has a different perimeter from the other three and to explain how they can tell that the other three shapes have the same perimeter. For the first set all of the shapes have a perimeter of 16 units except shape B. For the second set all of the shapes have a perimeter equal to the perimeter of the circle (3 units) except shape A. For the third set all of the shapes have a perimeter of 14 units except shape A. A A B C

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