06 AQA MOD F2_F082-088.qxd:Layout 1
6
27/5/10
15:39
Page 36
Geometry: Perimeter and area
6.1 Perimeter HOMEWORK 6A
G
1
Find the perimeter of each of the following shapes. Draw them on centimetre-squared paper first if it helps you. 8 cm 5 cm a b c 6 cm 3 cm 10 cm
5 cm
d
e
2 cm
2 cm
f
2 cm
5 cm 2 cm
2 cm 3 cm 5 cm
2 cm 2 cm
10 cm
6 cm
5 cm
2 cm 5 cm
4 cm
F
2
Draw as many different rectangles as possible that have a perimeter of 14 cm.
PS
3
Is it possible to draw a rectangle with a perimeter of 9 cm? Explain your answer.
AU
4
Which shape is the odd one out? Give a reason for your answer. 5 cm 3 cm
FM
5
4 cm 1 cm
A
7 cm B
3 cm
C
Simon wants to put a fence around three sides of a lawn. How much fencing does he need? 6m
4m
36
UNIT 3
FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving
06 AQA MOD F2_F082-088.qxd:Layout 1
27/5/10
15:39
Page 37
CHAPTER 6: Geometry: Perimeter and area
6.2 Area of an irregular shape HOMEWORK 6B 1
By counting squares, find the area of each of these shapes, giving answers in cm2. a b
c
G
d
By counting squares, estimate the area of each of these shapes, giving answers in cm2.
2
a
b
c
d
UNIT 3
37
06 AQA MOD F2_F082-088.qxd:Layout 1
28/5/10
06:32
Page 38
CHAPTER 6: Geometry: Perimeter and area
G
FM
3
Mr Bluegum, a forester, needs to find an estimate for the area of a forest. On the map below, the forest is shown in green.
Each square on the map represents 1 km2. Find an estimate for the area of the forest for Mr Bluegum. PS
4
This shape is drawn on a centimetre-squared grid.
a b
38
UNIT 3
Write down the area of the shape. On centimetre-squared paper, draw a square that has the same perimeter as the shape.
06 AQA MOD F2_F082-088.qxd:Layout 1
27/5/10
15:39
Page 39
CHAPTER 6: Geometry: Perimeter and area
AU
5
George says that he can find an estimate for the area of a circle by first finding the area of a square around the circle and then finding the area of a square inside the circle. The answer is the value between these two numbers.
F
Show how George finds an estimate for the area of this circle.
6.3 Area of a rectangle HOMEWORK 6C 1
Calculate the area and the perimeter of each rectangle below. a b 5 cm 4 cm
c
F
2m
2 cm 4 cm 8m 12 mm
d
e
20 m
3 mm 10 m
2
Copy and complete the following table for rectangles a to e.
a b c d e 3
Length 4 cm 7 cm 6 cm
Width 2 cm 4 cm
Perimeter
E
Area
22 cm 3 cm 30 cm
15 cm2 50 cm2
A square has a perimeter of 24 cm. What is its area? UNIT 3
39
06 AQA MOD F2_F082-088.qxd:Layout 1
16/7/10
08:12
Page 40
CHAPTER 6: Geometry: Perimeter and area
D
Copy and complete the following. a i 1 cm2 = .......... mm2 ii 3 cm2 = .......... mm2 2 2 b i 1 m = .......... cm ii 4 m2 = .......... cm2
4
PS
iii 12 cm2 = .......... mm2 iii 10 m2 = .......... cm2
This shape is made from four rectangles that are all the same size. Work out the area of one of the rectangles.
5
12 cm
FM
The diagrams show the size of Lin’s kitchen wall and the size of the square tile she wants to use to tile the wall. They are not drawn to scale.
6
20 cm 3m
20 cm
5m
What is the minimum number of tiles Lin will need to cover the wall? Remember to change the measurements of the wall into centimetres first.
6.4 Area of a compound shape HOMEWORK 6D Calculate the area of each shape below.
1
D
a
b
3 cm
c
8 cm 2 cm 2 cm
8 cm
6 cm 3 cm 6 cm
40
UNIT 3
4 cm
5 cm 2 cm
5 cm 2 cm
5 cm 2 cm
06 AQA MOD F2_F082-088.qxd:Layout 1
16/7/10
08:15
Page 41
CHAPTER 6: Geometry: Perimeter and area
d
10 cm
e
5 cm
D
10 cm 6 cm
10 cm
3 cm
10 cm 10 cm 6 cm
10 cm
5 cm
FM
2
Mr Jackson is fixing Formica® onto a worktop in his kitchen. Formica® comes in rolls 5 metres long and 0.5 metres wide. This is a sketch of the worktop. 3m 0.5 m 2.5 m
0.5 m
a b AU
3
Work out the area of the worktop. Does he have enough Formica® in one roll to cover his worktop?
Rachael says that the area of this shape is 64 cm2. Is she correct? Give a reason for your answer.
4 cm
8 cm 10 cm
2 cm 10 cm
PS
4
This L-shape is made from two rectangles that are the same size. It has an area of 48 cm2.
10 cm
Find the length and width of each rectangle. UNIT 3
41
06 AQA MOD F2_F082-088.qxd:Layout 1
27/5/10
15:39
Page 42
CHAPTER 6: Geometry: Perimeter and area
6.5 Area of a triangle HOMEWORK 6E Example
Find the area of this triangle Area = 1–2 × 7 × 4
4 cm
= × 28 = 14 cm 1– 2
2
7 cm
D
1
Write down the perimeter and area of each triangle. a b 5 cm
3 cm
10 cm
4 cm 2
20 cm
c
8 cm
21 cm
29 cm
6 cm
Work out the area of each of these compound shapes, made from rectangles and right-angled triangles. a b c
4m
8m
7m
20 m 12 cm
4m
8m
8 cm
15 m
4m
4 cm 3
Find the area of the wood on this blackboard 90° set square. 30 cm
24 cm 24 cm 30 cm
4
Which of these three triangles has the smallest area? a b 6 cm
c
5 cm 9 cm
12 cm
10 cm
4 cm
42
UNIT 3
06 AQA MOD F2_F082-088.qxd:Layout 1
27/5/10
15:39
Page 43
CHAPTER 6: Geometry: Perimeter and area
AU
5
D
Jen and Jack are comparing their answers to this question. Work out the area of this right-angled triangle.
10 cm
6 cm
8 cm Jen’s answer
Jack’s answer
A 8 6 46 24 cm2
A 8 10 4 10 40 cm2
Who is correct? Give a reason for your answer. PS
6
Work out the area of this rhombus.
20 cm
12 cm
The diagonals of the rhombus intersect at right angles.
UNIT 3
43
06 AQA MOD F2_F082-088.qxd:Layout 1
27/5/10
15:39
Page 44
CHAPTER 6: Geometry: Perimeter and area
HOMEWORK 6F Example
Find the area of this triangle. Area = 1–2 × 9 × 4
4 cm
= × 36 = 18 cm 1– 2
2
9 cm
D
1
Calculate the area of each of these triangles. a b
c 7 cm
5 cm
28 cm
10 cm
8 cm
22 cm
d
e
5m
f
3m 12 cm
10 cm
20 cm
9 cm
2
Copy and complete the following table for triangles a to e. a b c d e
3
Base 6 cm 10 cm 5 cm 4 cm
Vertical height 8 cm 7 cm 5 cm
Area
12 cm2 50 cm2
20 cm
Find the area of each of the shaded shapes. a b 40 cm
c
10 cm 8 cm 12 cm 80 cm
6 cm
7 cm
6 cm 8 cm
8 cm 12 cm
4
44
UNIT 3
Draw diagrams to show two different-sized triangles that have the same area of 40 cm2.
06 AQA MOD F2_F082-088.qxd:Layout 1
27/5/10
15:39
Page 45
CHAPTER 6: Geometry: Perimeter and area
PS
5
D
The rectangle and triangle below have the same area.
16 cm 4 cm
3 cm
Work out the length of the base of the triangle. AU
6
C
What is the same and what is different about these two triangles? 5 cm
8 cm
5 cm 8 cm
FM
7
Mary is making a mosaic from coloured tiles.
Each triangle has the following measurements:
5 cm
5 cm
How many tiles does Mary need to cover this rectangular board completely without leaving any gaps? 25 cm
20 cm
UNIT 3
45
06 AQA MOD F2_F082-088.qxd:Layout 1
27/5/10
15:39
Page 46
CHAPTER 6: Geometry: Perimeter and area
6.6 Area of a parallelogram HOMEWORK 6G Example
Find the area of this parallelogram. Area = 8 × 6 = 48 cm2
6 cm
8 cm
D
1
Calculate the area of each parallelogram below. a b 3 cm
5 cm
5 cm 8 cm c
d
4m 4m
10 cm 24 cm
2
Find the area of the shaded section. 8 cm 8 cm 8 cm 8 cm
20 cm
46
UNIT 3
06 AQA MOD F2_F082-088.qxd:Layout 1
27/5/10
15:39
Page 47
CHAPTER 6: Geometry: Perimeter and area
AU
3
D
Which two shapes have the same area? Show your working. a
b
c
7 cm 6 cm
5 cm
4 cm 9 cm
12 cm
PS
4
A square has the same area as this parallelogram. 4.5 cm 8 cm
What is the perimeter of the square?
6.7 Area of a trapezium HOMEWORK 6H 1
D
Calculate the perimeter and the area of each of these trapeziums. a 5 cm 5 cm
Be careful not to use the slanting side as the height.
4.1 cm
4 cm
9 cm
b
6 cm
7 cm
10 cm
13 cm
2
Calculate the area of each of these shapes. a 7m
b
10 cm
4m
3 cm 5 cm
3m
3m 2 cm 15 m 8 cm
UNIT 3
47
06 AQA MOD F2_F082-088.qxd:Layout 1
16/7/10
08:18
Page 48
CHAPTER 6: Geometry: Perimeter and area
D
3
Calculate the area of the shaded part in each of these diagrams. 6 cm a b 4 cm 4 cm
8 cm 5 cm
3 cm
6 cm 2 cm
2 cm
5 cm
9 cm 4
Which of the following shapes has the larger area? a b 3 cm
8 cm 2.4 cm
2.5 cm
5 cm 5
Priya is writing down her solution to this question. Work out the area of this trapezium. 10 cm 5 cm
16 cm
This is her answer. Area
(10 16) 5 (5 16) 5 21 5
She has made two mistakes. Write out a correct solution to the question. 6
The side of a swimming pool is a trapezium, as shown in the diagram. Calculate its area. 25 m 1.5 m 4m
48
UNIT 3
06 AQA MOD F2_F082-088.qxd:Layout 1
27/5/10
15:39
Page 49
CHAPTER 6: Geometry: Perimeter and area
PS
7
D
The area of this trapezium is 40 cm2. Work out possible values for a and b. a 10 cm b
Problem-solving Activity Pick’s Theorem Map makers and surveyors often need to calculate complex areas of land. Here is a way to find the area of shapes drawn on a square dotty grid.
This quadrilateral has an area of 161–2 square units. The perimeter of the quadrilateral passes through nine dots. Thirteen dots are contained within the perimeter of the quadrilateral. Draw some quadrilaterals of different shapes and sizes on dotty paper. Make sure the vertices are all on dots on the paper. Investigate the connection between the area and the total number of dots inside and the total number of dots on the perimeter of the shape. Then, from your findings, write down Pick’s Theorem.
UNIT 3
49
Answers: New GCSE Maths AQA Modular Homework Book Foundation 2
Chapter 6 Geometry: Perimeter and area 6.1 Perimeter HOMEWORK 6A 1 a 20 cm b 18 cm c 36 cm d 18 cm e 32 cm f 36 cm 2 Examples of rectangles with perimeters of 14 cm (1 × 6, 2 × 5, 3 × 4) 3 Yes, use fractions of a cm, e.g. a rectangle 2 cm by 2.5 cm. 4 C: the other two both have a perimeter of 16 cm. 5 16 m
6.2 Area of an irregular shape HOMEWORK 6B 1 2 3 4 5
a
6 cm2
b
13 cm2
c
1
4 2 cm2
d 5 cm2 a 9–11 cm2 b 11–13 cm2 c 13–15 cm2 d 12–14 cm2 15–18 km2 a 7 cm2 b 4 by 4 square 2 Outer area = 36 cm ; inner area 16 cm2: (36 + 16) ÷ 2 = 26 cm2
6.3 Area of a rectangle HOMEWORK 6C 1 a 10 cm2, 14 cm d 36 mm2, 30 mm 2 a 12 cm, 8 cm2 d 5 cm, 16 cm 3 36 cm2 4 a i 100 b i 10 000 2 5 48 cm 6 375
b e b e
16 cm2, 16 cm c 16 m2, 20 m 200 m2 , 60 m 22 cm, 28 cm2 c 5 cm, 30 cm2 10 cm, 5 cm or 5 cm, 10 cm
ii ii
300 40 000
iii 1200 iii 100 000
6.4 Area of a compound shape HOMEWORK 6D 1 a 33 cm2 b 40 cm2 d 60 cm2 e 500 cm2 2 a 2.5 m2 b Yes, the area in one roll is 2.5 m2 3 She is not correct. The area is 52 cm2. 4 6 cm and 4 cm
© HarperCollinsPublishers 2010
c
60 cm2
14
Answers: New GCSE Maths AQA Modular Homework Book Foundation 2
6.5 Area of a triangle HOMEWORK 6E b 24 cm, 24 cm2 c 70 cm, 210 cm2 1 a 12 cm, 6 cm2 2 a 40 cm2 b 168 m2 c 32 m2 2 3 162 cm 4 c: 24 cm2 5 Jen, as she used the correct height; Jack used the slanting side. 6 120 cm2 HOMEWORK 6F b 35 cm2 c 308 cm2 1 a 20 cm2 e 54 cm2 f 100 cm2 d 7.5 m2 2 a 24 cm2 b 35 cm2 c 12.5 cm2 d 6 cm e 5 cm 2 3 a 1800 cm b 144 cm2 c 116 cm2 4 Students should have drawn two triangles with the product of base and height 80 cm2. 5 3 cm 6 Areas are the same but the perimeters are different. 7 40
6.6 Area of a parallelogram HOMEWORK 6G 1 a 15 cm2 d 240 cm2 2 256 cm2 3
b and c;
4
24 cm
1 2
b
40 cm2
c
16 m2
× 12 × 6 = 36 cm2 and 9 × 4 = 36 cm2
6.7 Area of a trapezium HOMEWORK 6H b 36 cm, 66.5 cm2 1 a 23.1 cm, 28 cm2 2 a 89 m2 b 35.5 cm2 b 24 cm2 3 a 45 cm2 4 a is larger (a is 10 cm2 and b is 9.6 cm2) 5 Incorrect multiplication of terms inside brackets (she should have multiplied both terms by 12 ) and units are incorrect; correct answer is 65 cm2. 6 7
68.75 m2 a + b = 8 with a < b
© HarperCollinsPublishers 2010
15
Answers: New GCSE Maths AQA Modular Homework Book Foundation 2
Problem-solving Activity: Pick’s theorem Pick's theorem provides a simple formula for calculating the area, A, of a polygon constructed on a grid of equally spaced points, so that each vertex is located on one of the grid points. This would be like every vertex falling on the grid lines of a coordinate graph so that they all have integer coordinates. If the number of points that fall inside the polygon is taken as i and the number of points that are located exactly on the perimeter of the polygon is b, then the area is given by: b A = i + −1 2
© HarperCollinsPublishers 2010
16
