Areas of Shapes
Area of a Rectangle
Area
= length × width
= a×b
Example 1
Find the area of the rectangle:
Area
Area of a Square
A square is a rectangle with equal sides.
Area
= length × width
=a×a
= a2
Example 2
Find the area of the square:
Area = 3 cm × 3 cm
= 9 cm2
Gareth Lotwick, March 2006
= 5 mm × 3 mm
= 15 mm2
Calculating the Area of a Triangle
Case 1: Right-angled Triangles
It can be seen that the area of ∆ ABC is half the area
of the rectangle ABCD.
Area of ∆ ABC = ½(AB × BC)
= ½ base length × perpendicular height
Case 2: Triangles which are not right angled
Consider a ∆ ABC as shown which is not rightangled.
Draw AD such that AD is perpendicular to BC
Area of ∆ ABC = Area ∆ ABD + Area ∆ ACD
= ½(BD×AD) + ½(DC×AD)
= ½ (BD + DC)×AD
= ½ BC×AD
= ½ base length × perpendicular height
So whether a triangle is right-angled or not, the area can be calculated in the same way.
Area of a Triangle
Area
= ½ base length × perpendicular height
= ½ a ×b
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Example 3
Find the area of ∆ ABC
Area = ½ × 4cm × 5cm
= 10 cm2
Example 4
Find the area of ∆ ABC
Area = ½ × 4 cm × 6 cm
= 12 cm2
Area of a Trapezium
This is a quadrilateral with one pair of parallel sides.
Area = ½ ×sum of parallel sides × distance between them
= ½ (a + b) c
So why is this the area of a trapezium?
Area ABCF = Area ∆ AEF + Area ABDE + Area ∆ BDC
= ½ EF×c + ac + ½ CD×c
= [ ½EF + a + ½ CD ] c
= [ ½a + (½EF + ½a + ½ CD) ] c
= ½ [ a + (EF + a + CD)] c
= ½ (a + b) c
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Example 5
Find the area of the Trapezium
Area
= ½ (12 mm + 21 mm) × 8 mm
= ½ × 33 × 8 = 132 mm2
Area of a Circle
Area = π r2
where r is the radius of the circle.
Area of a Semi-circle
This is half the area of a circle, so
Area = ½ π r 2
where r is the radius of the circle.
Example 6
Find the area of the circle
Area = π × 52
= 78.54 cm2
Example 7
Find the area of the semi-circle
Radius = ½ × 3 mm
= 1.5 mm
Area = ½ π × 1.52
= 3.53 mm2
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Example 8
A garden consists of a rectangular lawn and semi-circular flower bed of radius 2.5 m as
shown. Determine the area of the garden.
Area of lawn = 8 × 5 = 40 m2
Area of flower bed =½ π × 2.52 = 9.82 m2
Total area of garden = 40 + 9.82 = 49.82 m2
Exercises
1. Find the area of a rectangle, length 12 m, breadth 8 m.
2. Find the area of the triangle shown
3. A triangle with a base of 3m has an area of 30m2. What is the height of the triangle?
4. Find the area of the trapezium shown
5. Find the area of a circle with diameter 4 mm.
6. Find the area of a semi-circle of radius 3 cm.
7. A circle has an area of 100m2, what is its radius?
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Answers
1) 96m2
2) 15cm2
3) Area ∆ = ½ base length × perpendicular height
30m2 = ½ × 3m × height
= 1.5 × height
So height = 30 / 1.5 = 20m
4) 16.5m2
5) 12.57 mm2
6) 14.14cm2
7) Area = π r 2
100m2 = π r 2
100
So r =
= 5.64 m

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