diameter radius circumference The perimeter of a circle is called the circumference (C). The...
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Transcript of diameter radius circumference The perimeter of a circle is called the circumference (C). The...
AREA AND CIRCUMFERENCE OF A CIRCLE
diam
eter
radius
circumference
The perimeter of a circle is called the circumference (C).
The diameter (d) of a circle is twice the radius (r).
C 2r or C d
3.142
rearranging the sectors of the circle gives
Consider dividing a circle into 8 equal sectors:
Now consider dividing the circle into 12 equal sectors:
rearranging the sectors of the circle gives
The more sectors there are, the closer the area gets to being a rectangle.
r
rhalf circumference
Area r 2
Examples1 Calculate a the circumference b the area of the circle.
8 cm
a Circumference d
8
25.1 cm (to 3 s.f.)
b Area r 2
42
50.3 cm2 (to 3 s.f.)
16
Examples2 Calculate a the circumference b the area of the circle.
3.7 cm
a Circumference d
7.4
23.2 cm (to 3 s.f.)
b Area r 2
3.72
43.0 cm2 (to 3 s.f.)
13.69
Examples3 The radius of the circle is 5 cm. The circle touches all four sides of the square. Calculate the shaded area.
Shaded area area of square area of circle
10 10 52
21.5 cm2 (to 3 s.f.)
100 25
10 cm
Examples4 The radius of the large circle is 8 cm. The radius of the small circle is 5 cm. Calculate the shaded area.
Shaded area area of large circle area of small circle
82 52
126 cm2 (to 3 s.f.)
64 25
39
Examples5 The radius of the circle is 3 cm. The vertices of the square lie on the circumference of the circle. Calculate the shaded area.
Let length of sides of square x
x2 x2 62
x 4.243... cm
2x2 366 cm
Shaded area area of circle area of square
32 4.243 4.243
10.3 cm2 (to 3 s.f.)
9 18
x2 18
x
x
Using Pythagoras
Examples6 The area of the circle is 200cm2. Find the value of r.
r cm Area r 2
200 r 2
r 63.66
r 2
200
r2 63.66
r 7.98 cm (to 3 s.f.)
Semicircles and quadrants
A semicircle is half a circle. A quadrant is quarter of a circle.
Examples1 Calculate a the perimeter b the area of the semicircle.
10 cm
a Perimeter
d
210
10
210
25.7 cm (to 3 s.f.)
b Area
r 2
2
52
2
39.3 cm2 (to 3 s.f.)
Examples2 Calculate a the perimeter b the area of the quadrant.
9 cm
a Perimeter
d
4 9 9
18
218
46.3 cm (to 3 s.f.)
b Area
r 2
4
92
4
63.6 cm2 (to 3 s.f.)
Examples3 The perimeter of the quadrant is 50 cm. Calculate the value of r.
r cm
Perimeter
2r
4 r r
50
r
2 2r
50 1.571r 2r
50 3.571r
r 14.0 cm (to 3 s.f.)
r
50
3.571
Examples4 The diagram shows four identical circles of radius 4 cm. Calculate the shaded area.
Shaded area area of square area of 4 quadrants
8 8 42
13.7 cm2 (to 3 s.f.)
64 16
area of square area of circle
Arcs and sectors
An arc is part of the circumference of a circle.
A sector of a circle is a region bounded by an arc and two radii.
r
r
sector
arc
Arc length
360
2r
Area of sector
360
r 2
95o 5 cm
5 cm
O
A
B
Examples1 Find a the area of sector OAB
b the length of arc ABc the perimeter of sector OAB.
a Area of sector
95
360 52
20.7 cm2 (to 3 s.f.)
b Arc length
95
3602 5
8.2903...
8.29 cm (to 3 s.f.)
c Perimeter OA OB arc AB
5 5 8.2903...
18.3 cm (to 3 s.f.)
Examples2 Find a the area
b the perimeter of the shape.
7 m 3 m
a Area
126
360 102
126
360 72
56.1 m2 (to 3 s.f.)
b Perimeter large arc small arc 3 3
21.99 15.34 3 3
43.4 cm (to 3 s.f.)
126° 110.0 53.88
126
3602 10
126
3602 7
3 3