Length, perimeter and area 3.1. Example

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1 3.1 Length, perimeter and area Kno and use the names and abbreviations for units of length and area Be able to measure and make a sensible estimate of length and area Find out and use the formula for the perimeter of a rectangle and kno ho to calculate the perimeter of shapes made from rectangles Key ords millimetre (mm) centimetre (cm) metre (m) kilometre (km) perimeter area Find out and use the formulae for the areas of a rectangle and of a right-angled triangle Solve problems in everyday life involving length and area Units of length are the millimetre (mm), centimetre (cm), metre (m) and kilometre (km). 10 mm 1cm 1000 mm 100 cm 1m mm cm 1000 m 1km Perimeter means the distance all the ay around the boundary of a shape. The perimeter of a rectangle is: 2 base 2 The area is the space inside a to-dimensional shape. Units of area are the square millimetre (mm 2 ), square centimetre (cm 2 ), square metre (m 2 ) and square kilometre (km 2 ). We can calculate the area of a rectangle by We can calculate the area of a right-angled multiplying the base by the. triangle by imagining it is half of a rectangle. Area base Area 1 2 (base ) base base Example a) Find the perimeter of this shape. b) Find the area of this shape. a) AB CD EF 6cm 4cm Perimeter AB BC CD DE EF FA 7cm 6cm 3cm 4cm 40 cm b) Area of rectangle X 7cm 70 cm 2 Area of rectangle Y 3cm 4cm 12 cm 2 A X D Y B 7 cm C Total area of shape 70 cm 2 12 cm 2 82 cm 2 F E A D E F Find the length AB first. B 7 cm C Here the shape has been broken don into to rectangles to find the area. An alternative method ould be to subtract the area of the rectangle that is missing from the corner of the large rectangle. 26 Maths Connect 1R

2 Exercise What units ould you use to measure a) the perimeters b) the areas of the folloing shapes: Choose from: mm, cm, m, km, mm 2, cm 2, m 2 and km 2. i) the sole of your foot ii) the cover of this book Change all the iii) the top of the table iv) the floor of your bedroom measurements to mm first. v) a mouse s footprint vi) Spain. Calculate the perimeters of these shapes. a) 9 m b) A 3 m B c) A B 3 m 6 m 1 m 2 cm 6 m D 3 m 2 m D 1 m C C 15 m F mm E F E 32 mm These shapes are made from rectangles. Find their areas. a) 5 m b) c) 1 m 1 m 1 m 7 m H 4. G 6 m 1 m 2 m 1 m 15 m Find the areas of these right-angled triangles: a) b) c) mm Find the base and of rectangles hich have: a) area of 2, perimeter 1 b) area of 2 2, perimeter 22 cm c) area of 16 mm 2, perimeter 16 mm d) area of 7.5 km 2, perimeter 13km. Explain ho you orked these out. You might find it helpful to sketch the rectangles. Remember that a square is also a type of rectangle. Investigation Use squared paper. a) Ho many different rectangles can you dra using 12 hole squares? b) Do they all have the same perimeter? c) Do they all have the same area? Make other shapes using 12 hole squares and calculate their perimeters and areas. d) What do you notice? e) Write a hint for someone ho is trying to make a rectangle from 100 squares ith the smallest possible perimeter. Length, perimeter and area 27

3 3.2 Area of composite shapes Kno ho to calculate the areas of a triangle, a parallelogram and a trapezium Key ords square centimetre (cm 2 ) square metre (m 2 ) square millimetre (mm 2 ) square kilometre (km 2 ) area The area is the space inside a to-dimensional shape. The formula for the area of a rectangle is base The formula for the area of a right-angled triangle is 1 2 (base ) Shapes that do not have right angles have slant s as ell as perpendicular s. We need to kno the perpendicular of the shape to find the area. slant perpendicular We can calculate the area of a parallelogram by doubling the area of a non right-angled triangle. Area base (perpendicular) perpendicular base These triangles have the same area. We can calculate the area of a non- rightangled triangle by splitting it into to right-angled triangles, or by imagining it is half a rectangle. Area 1 2 (base perpendicular ) We can find the area of a trapezium by breaking it into a rectangle and triangles and then calculating the areas of these shapes. perpendicular base These triangles have the same area. Example Find the area of this shape. Area of parallelogram base perpendicular 6cm 4cm 2 2 Area of triangle 1 2 (base perpendicular ) (10 4) Total area The shape can be broken don into a parallelogram and a triangle. 42 cm 2 28 Maths Connect 1R

4 Exercise Calculate the areas of these triangles: a) 5 mm b) c) 32 mm 1 43 mm Calculate the areas of the folloing shapes: a) b) c) d) 4 m 15 m 7 m 7 cm 2 cm 2 cm The shaded shapes on this grid can be broken don into simpler shapes. Calculate the areas, in square units, of each of the shaded shapes. a) b) c) Here are to trapeziums ith parallel sides of length and and a perpendicular of 2 cm: 2 cm 1. 2 cm 0. a) Find the areas of the to trapeziums. b) Dra to more trapeziums ith the same dimensions and find their areas. What do you notice? Investigation Albert has a rabbit and 12 m of fencing. He ants to make a run for the rabbit. Albert thinks that the area enclosed by the fencing is alays the same, regardless of the shape of the run. Sho hy Albert is rong. What is the largest area he could enclose? Sho ho you kno. Area of composite shapes 29

5 3.3 Connecting 2-D and 3-D Use other 2-D shapes to visualise and describe 3-D shapes and consider their properties Be able to dra 2-D representations of 3-D shapes Key ords face edge vertex vertices cube cuboid 3-D A face is the flat surface of a solid. An edge is here to faces meet. A vertex Vertices is here three or more edges meet. is the plural of vertex. vertex face edge A cube has six identical square faces. A cuboid has three pairs of rectangular faces. Opposite faces are the same shape and size. A tetrahedron has four triangular faces. A prism has a uniform cross section. A hemisphere has one circular face and one curved face. A cylinder has to circular faces and one curved face. cube cuboid tetrahedron triangular prism hemisphere cylinder We sometimes use isometric paper to dra representations of 3-D shapes. cube Example 1 Describe this shape. This shape is a prism. It has to L-shaped faces and six rectangular faces. It has 18 edges and 12 vertices. Example 2 Dra this shape on isometric paper. Dra all of the front faces first, then the side faces and finally the top faces. 30 Maths Connect 1R

6 Exercise Work ith a partner. Take a handful of cubes each. Take it in turns to make a 3-D shape and describe it so that your partner can make it. If your cubes are coloured, try not to use colour as a clue! Use isometric paper to dra these shapes made from cubes. Shade the front faces. a) b) c) d) Build a shape game. You ill need a normal dice and some isometric paper. Take turns to thro the dice. 1 square; 2 rectangle; 3 triangle; 4 circle; 5 hexagon; 6 another thro. Collect a face for each number you thro. The inner is the first to collect enough 2-D faces to make a 3-D shape and to sketch that shape. This skeleton 3-D shape is made from four lengths of ire and eight lengths of ire. Write the list of lengths of ire for each of the skeletons shon belo: M N O Dra all the front faces first. P Q R 12 cm a) For each of the shapes in Q4, record the number of vertices, face and edges. Can you find the connection beteen the number of vertices, faces and edges in each shape? b) Ho many edges ould a shape ith eight faces and telve vertices have? Sketch a shape like this on isometric paper. Try adding the number of faces and vertices together. Investigation Do not count There is only one possible model that can be made from to cubes. rotations of the There are only to possible models that can be made from three cubes. same model. Make models from four and five cubes and dra them on isometric paper. Ho can you be certain that you have made all the possible models? Connecting 2-D and 3-D 31

7 3.4 Cubes and cuboids Investigate the nets of cubes and cuboids Kno and use a formula for the surface area of a cube and cuboid Solve simple problems involving lengths, perimeters and surface areas of shapes made from cubes and cuboids Key ords surface area net cube cuboid The surface area of a 3-D shape is the total area of all its faces. Draing the net helps us to make sure e have added the areas of all of the faces. The surface area of a cube of side is the sum of the area of its six faces. This is calculated as 6 (5 5) cm 2. The general formula for the surface area of a cube of side is 6 ( ). The surface area of a cuboid of idth, length and is the sum of the areas of its three pairs of faces. This is calculated as h 2 (3 4) 2 (3 5) 2 (4 5) cm 2. The general formula for the surface area of a cuboid of idth, length l and h is l l 2 (l ) 2 (l h) 2 ( h). h h h Example Find the surface area of these shapes. a) b) a) Area of purple rectangle 6 60 cm 2 Area of pink rectangle 3 30 cm 2 Area of bron rectangle Total surface area b) Area of yello face but there are 2 of these so area of yello faces 72 Area of the 2 green faces Area of the 2 blue faces Area of the base Area of the back Total surface area square units Alternatively, you can use the formula for surface area, given in the explanation box. Replace, l and h ith the idth, length and of the cuboid. You may find it helpful to sketch the different faces of the shape ith their dimensions. 32 Maths Connect 1R

8 Exercise Here are the nets of different cuboids. For each one, find the area of the yello, blue and green faces. Then find the total surface area for each net. a) 3 mm 10 mm 3 mm b) c) 1 m 0.1 m d) 5 mm 5 mm 2 m 3 mm 70cm 5 mm 2 m 3 mm 0.1 m 70 cm Find the surface areas of each of the folloing shapes: a) b) c) d) 1 m 30 cm 1 These shapes are made from cuboids. a) Find the areas of the front faces. b) Find the total surface areas of the shapes. 2 cm 7 cm 9 cm Mr Harding is building a rectangular pond for his garden. The pond is 1 m deep. The length is 2 m and the idth is 2.5 m. He ants to line the sides and bottom of his pond ith tiles that are 2 square. Ho many tiles ill he need? Investigation You have to of each of the folloing rectangles: Rectangle Rectangle Rectangle Rectangle Rectangle Rectangle Rectangle Rectangle Length (cm) Width (cm) Investigate all the different cuboids that can be made using these rectangles as faces. Keep a record of the length, idth, and surface area of the different cuboids. What do you notice? Cubes and cuboids 33