4.04 Area

Worksheet
Areas of squares and rectangles
1

Find the area of the following rectangles:

a
b
c
2

Find the area of the following squares:

a
b
3

Find the area, A, of a rectangle when l is 14 \text{ cm} and w is 5 \text{ cm} using the formula A = l \times w.

4

Find the area of a rectangle whose length is 12 \text{ cm} and width is 5 \text{ cm}.

5

Find the area of a square whose side is 5 \text{ cm}.

6

Find the area of a square whose perimeter is 8\text{ cm}.

7

The rectangle shown has side lengths given in millimetres.

a

State the dimensions of the rectangle in centimetres:

i

Length

ii

Width

b

Hence, find the area of the rectangle in square centimetres.

8

Calculate the area of the rectangle shown in square metres:

9

A rectangle measures 390 \text{ cm} wide by 12.5 \text{ m} long. Calculate the area of the rectangle in square metres.

10

A rectangle has side lengths of 700 \text{ m} and 2900 \text{ m}. Calculate the area of the rectangle in square kilometres.

Side lengths from areas
11

Find the width of the rectangle shown with an area of 24\text{ m}^2 and a length of 8\text{ m}:

12

Find the side length of a square with an area of 25\text{ mm}^2:

13

Find the length of a rectangle that has an area of 40\text{ cm}^2 and a width of 5 \text{ cm}.

14

Find the width of a rectangle that has an area of 27\text{ mm}^2 and a length of 9 \text{ mm}.

15

Find the length of each side of the square whose area is 64\text{ cm}^2.

16

Find the perimeter of a square whose area is 49\text{ cm}^2.

Area of triangles
17

For each of the following figures:

i

Find the area of the entire rectangle.

ii

Find the area of the shaded triangle.

a
b
c
d
18

Find the area of the following triangles:

a
b
c
d
e
19

Calculate the area of the triangle shown in square centimetres.

20

Find the area of the following triangles:

a

A triangle whose base, b, is 3 \text{ m} and height, h, is 2 \text{ m}.

b

A triangle whose base b is 6\text{ m} and height h is 12\text{ m}.

Dimensions of triangles
21

Find the base of the triangle whose area is 17.5\text{ cm}^2 and height is 5 \text{ cm}.

22

Find the value of the pronumeral for each of the following triangles, given the area:

a

The area of this triangle is 54\text{ cm}^2:

b

The area of this triangle is 120 \text{ cm}^2.

c

The area of this triangle is 24 \text{ m}^2.

d

The area of this triangle is 120 \text{ mm}^2.

e

The area of this triangle is 32 \text{ cm}^2.

f

The area of this triangle is 12 \text{ mm}^2.

23

The following table shows measurements for three different triangles that all have an area of 42\text{ cm}^2:

Find the value of:

a
A
b
B
c
C
24

Give three examples of pairs of values that could be the dimensions, base and height, of a triangle with an area of 24 \text{ cm}^2.

Area of composite shapes
25

Consider the following figure:

a

Find the area of rectangle B.

b

Find the total area of the given shape.

26

For the following composite shape:

a

Find the area of rectangle A.

b

Find the total area of the given shape.

27

The quadrilateral in the diagram has a diagonal measuring 14\text{ m}.

Calculate the total area of the quadrilateral.

28

Find the area of the following composite shapes:

a
b
c
d
e
f
g
h
i
j
k
l
29

Consider the following figures, where all measurements are in centimetres:

• Figure 1:

• Figure 3:

• Figure 2:

• Figure 4:

a

Find the area of the entire rectangle shaded as shown in figure 1.

b

Find the area of the interior rectangle shaded as shown in figure 2.

c

Find the area of the corner rectangle shaded as shown in figure 3.

d

Hence, calculate the area of the shaded region as shown in figure 4.

30

Find the area of the shaded region in the following diagrams:

a
b
c
d
e
f
31

Consider the following figure:

a

Find the value of x.

b

Find the area of the shaded region.

Applications
32

A kitchen floor is tiled with the tiles shown in the picture. If 30 tiles are needed to tile the floor, find the total area of the floor.

33

A window containing a rectangular piece of glass is broken and needs replacing. Emma is told that it will cost 5 cents per square metre to replace. She measures the window to have a width of 27 \text{ cm} and a height of 15 \text{ cm} to get an idea of the price. When the job is completed, the repairman bills Emma for \$35.65, and explains that there is more glass that cannot be seen since some of the glass sits within the frame: a How much did Emma expect the glass replacement to cost? b What was the area of the piece of glass that was actually replaced? c If the actual glass extends 2 \text{ cm} on either end beyond the original width Emma measured, calculate the actual height of the window. 34 A rectangular driveway is to be resurfaced with gravel. A 3 cubic metre load of gravel will cover 12 square metres of driveway, and costs \$6.

A scale drawing of the rectangular driveway is given. The actual length of the longest side of the driveway is 16 metres.

a

Determine the actual width of the driveway.

b

How many 3 cubic metre loads of gravel will be required to resurface the entire driveway?

c

Calculate the total cost of the gravel.

35

A crop farmer trades 5 identical pieces of machinery, which have a market value of \$880\,000 each, for a square piece of land as shown in the scale drawing. The scale of the drawing is 1:3000. a Find the actual area of the square piece of land, in square metres. b Determine the total market value of the 5 pieces of machinery. c According to this exchange, find the value of the land per square metre, to the nearest dollar. 36 John is tiling a room floor that has a total area of 9\text{ m}^2. The tiles he is using are squares, measuring 25\text{ cm} by 25\text{ cm}. a Calculate the area of a single tile in \text{m}^2. b Calculate the number of tiles John will require to cover the entire floor area. 37 A rectangular garden bed measures 430\text{ cm} by 250\text{ cm}. A bag of fertiliser covers an area of 2 \text{ m}^2. a Calculate the number of whole bags of fertiliser needed to cover the total area of the garden bed. b Calculate the area in \text{m}^2 that the left-over fertiliser will be able to cover. 38 A national park has a rectangular flower bed that is 11\text{ m} long and 12 \text{ m} wide. Fertiliser costs \$4.33 per square metre.

a

Calculate the area of the flower bed.

b

Hence, calculate the cost of fertilising the flower bed.

39

A farmer wants to cover a rectangular section of roofing that measures 5\frac{1}{2} \text{ m} by 4\frac{1}{2} \text{ m} with solar panels. Having received quotes from various solar panel suppliers, she estimates that the panels will cost \$300 per square metre to install. Calculate the estimated cost of covering the section of roofing with solar panels. 40 Luke made a square mosaic that has side lengths of 3 metres. Luke decided to add a border to his mosaic, and now it has side lengths of 3.2 metres. By how much has the area of the mosaic increased? 41 A pizza maker is experimenting with different shaped pizza bases, but wants the perimeter of the base to stay at 78\text{ cm}. a If the rectangular pizza base is 30 \text { cm} long, what must be the width of the base? b What would be the area of this base? c If he changes the base so that one side measures 6 \text { cm} but the perimeter stays the same, how much less space will the base have for toppings? 42 A family has 7.2 \text{ m}^2 of kitchen floor space for a rectangular bench. a If the width is fixed at 2 metres, and the bench is to take up all 7.2 \text{ m}^2, find the required length of the bench. b The kitchen bench will be made up of bench top pieces that measure 0.5 metres in width and 1.2 metres in length. Find the maximum number of bench top pieces that can be fitted along the width of the bench. c How many pieces of bench top pieces will be needed in total to create the bench top? d The kitchen floor is to be covered with tiles whose dimensions will be \dfrac{1}{8} the width of the bench, and \dfrac{1}{20} the length of the bench. How many tiles will be covered by the bench? 43 The faces on a 4 sided die are all triangular. Each face has a base length of 13 \text{ mm} and a perpendicular height of 20 \text{ mm}. Find the area of one face of the die. 44 Sharon has purchased a rectangular piece of fabric measuring 12 \text{ m} in length and 7 \text{ m} in width. What is the area of the largest triangular piece she can cut out from it? 45 A gutter running along the roof of a house has a cross-section in the shape of a triangle. If the area of the cross-section is 50 \text{ cm}^2, and the length of the base of the gutter is 10 \text{ cm}, find the perpendicular height h of the gutter in metres. 46 At the entrance of the Louvre museum is a glass structure in the shape of a square base pyramid. A replica of this pyramid is to be built such that each triangular face of the pyramid measures 7.5 metres at the base, and has a perpendicular height of 10 metres. The faces will be made up of identical triangular glass tiles tessellated to fit exactly on each face. a Find the total area that needs to be covered with the triangular glass tiles. b How many glass tiles will be needed in total if each triangular tile measures 15 \text{ cm} across the base and has a perpendicular height of 20 \text{ cm}? 47 Deep sea divers are scanning an area of the sea bed where a boat capsized. They want to get to point P, which is h metres above the sea bed. At this point, they can cast a light out to view 9 metres across the sea bed and a cross sectional area of 45 \text{ m}^2 of water. From side-on, the light casts a shape that looks like the diagram below. The divers descend at a rate of 1.2 metres per second. a Find h, the distance of the divers from the sea bed at the point P. b If the divers were descending directly downwards, how high above the sea bed were they 6 seconds before they reached point P? 48 A sand pit set in the corner of a property has dimensions as shown below. a Calculate the area of the sandpit in square metres. b A 20 \text{ kg} bag of play sand costs \$7.80, and covers an area of 0.5\text{ m}^2 to an appropriate depth.

How much will it cost to buy enough bags of sand to fill this sand pit?

49

The lounge room floor shown is to be tiled with slate tiles. The cost of tiling the room is \\$25/\text{m}^2.

a

Find the area to be tiled.

b

How much will it cost to tile the room?

50

Katrina has a garden bed that is shaped as shown in the following figure:

Find the area of the garden bed.

51

Lachlan draws the plot of land which contains his house and garden.

a

Find the total area of the plot of land.

b

Find the area covered by the house.

52

A farmer is going to fertilise his paddock, which has the following shape:

a

Find the area of the paddock.

b

If it takes 150 kg of fertiliser to fertilise 100 square metres, how many kilograms of fertiliser should the farmer purchase?