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9
Perimeter and area
●●The tangram
aft
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The tangram is an ancient Chinese puzzle. The square is divided into
seven pieces and the pieces can be reassembled into other shapes.
You are going to use the tangram to revise some basic geometry
before moving on to the calculation of perimeters and areas of
triangles, quadrilaterals and circles.
Exercise 9.1
Make four copies of the tangram.
1 Cut out the pieces from one copy of the tangram. Rearrange the pieces to
make this shape.
Dr
Stick your solution into your exercise book.
2 Cut out the pieces from the second copy of the tangram.
Rearrange the pieces to make this shape.
Stick your solution into your exercise book.
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3 Cut out the pieces from the third copy of the tangram. Rearrange the pieces
to make up some designs of your own. Be as creative as you can.
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4 The original tangram set is arranged in a square, and one of the pieces is a
square itself. Show how you can make a square, using:
(a) two tangram pieces
(d) five tangram pieces
(b) three tangram pieces
(e) six tangram pieces.
(c) four tangram pieces
5 Show how you can use all seven tangram pieces to make:
(a) a trapezium
Remember that a
parallelogram has two
pairs of opposite sides
that are parallel.
(b) a rectangle that is not a square
(d) a triangle.
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(c)a parallelogram that is not a square
●●Tangrams and quadrilaterals
The original set of tangram pieces was arranged as a square of side 5 cm.
The area of this square is therefore 5 × 5 = 25 cm2.
What is the area of each of the shapes in Q 1–3 and Q5?
They are all 25 cm2 as well.
aft
Although you rearranged the pieces, the total area of all the pieces
did not change.
You will use this fact to discover some more area formulae.
Square
Rectangle
b
b
h
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Perimeter and area
First, here are some formulae that that you already know.
Area of square = b2
Area of rectangle = base × height = b × h
Perimeter of square = 4b
Perimeter of rectangle = 2b + 2h
The sides of a rectangle can be referred to as its length and width.
In this case, the formula would be:
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area of rectangle = length × width or l × w
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Tangrams and quadrilaterals
Now suppose you cut a triangle from one end of the rectangle and add
it to the other end.
Area = base x height
Area = base x height
The shape changes from a rectangle to a parallelogram. Even though
you have changed the shape, the area has stayed the same.
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Therefore, you should see that:
area of parallelogram = b × h
A rhombus is a parallelogram with four equal sides.
b
Using the same ideas as above:
b
area of rhombus = b × h
perimeter of rhombus = 4b
h
b
b
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When you divide a rectangle or a parallelogram in half you get two
identical triangles.
h
b
h
h
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b
So: area of a triangle =
1 b
2
b
×h
where the base and height are equal to the base and height of the
enclosing rectangle or parallelogram.
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For all of these shapes remember that the height must be
perpendicular to the base.
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base
base
height
Perpendicular
means ‘at right
angles to’.
height
base
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height
height
base
Exercise 9.2
Think again about the formula for the area of a triangle.
Work these out, given that b = 4 and h = 5
1 1 bh
3 b× h÷2
5 b× h
2 b ×h
4 b ×h
2
6 1 (b × h)
2
2
2
2
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●●Areas and perimeters of 2D shapes
You should have found that you get the same answer for the area of a
triangle every time. All the formulae mean the same.
It doesn’t matter which way you remember the formula but you will
usually see it written as:
area of a triangle = 1 b × h
To find the area or perimeter of a shape you follow the same steps as
when you substitute in any other algebraic formulae.
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Perimeter and area
2
1 Write the formula.
2 Substitute.
3 Calculate.
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4 Write the answer with the correct units.
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Areas and perimeters of 2D shapes
Example
(i) Calculate the area and perimeter of this rhombus.
= 7 × 5.5
5.5 cm
= 38.5 cm2
Perimeter of rhombus = 4b
7 cm
=4×7
= 28 cm
(ii) Calculate the area of this triangle.
Area of a triangle = 1 b × h
= 1 × 1.5 × 1.2
2
= 0.6 × 1.5
1.2 m
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2
Note that the units
are square units
because you are
calculating the area.
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Area of rhombus = b × h
1.5 m
It doesn’t matter which unit you
divide by 2 so choose the easier option.
= 0.9 m2
Exercise 9.3
1 Work out the perimeter of each shape.
8.7 mm
(c)
aft
(a)
12 mm
6.5 cm
(b)
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7m
12 m
2 Calculate the areas of the shapes in question 1
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3 Calculate the area of each shape.
(d)
8 cm
5 cm
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(a)
3m
2.4 m
10 cm
(b)
0.5 m
2.5 m
0.9 m
(e)
4m
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4m
0.3 m
3.5 m
0.8 m
(c)
4m
4m
40 cm
(f)
65 cm
0.5 m
1.2 m
1.3 m
0.5 m
aft
1.2 m
1.3 m
4 Calculate the area of each shape. All lengths are in centimetres.
(a)
(c)
7
5
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Perimeter and area
4
4
3
6
(b)
35
20
(d)
9
6
7
9
10
6
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