Transcript Unit 27 AREAS OF CIRCLES, SECTORS, SEGMENTS, AND ELLIPSES

Unit 27
AREAS OF CIRCLES,
SECTORS, SEGMENTS,
AND ELLIPSES
AREAS OF CIRCLES


The area of a circle is equal to the
product of  and the square of the
radius (A = r2)
The areas of two circles have the
same ratio as the squares of the radii
or diameters
2
2
A1
r1
d1
 2  2
A2
r2
d2
2
AREAS OF CIRCLES

The areas of two circles are 144 mm2
and 36 mm2. Compare the radius of
the larger circle with the radius of the
smaller circle:
A1
r12 r1
Since
 2;

A2
r2 r2
A1
A2

144
 2 Ans
36
– The radius of the larger circle is 2 times larger than
the radius of the smaller circle
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AREAS OF SECTORS


A sector of a circle is a figure formed
by two radii and the arc intercepted by
the radii
The area of a sector is given as:
 
θ
2
A
π
r
360
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AREAS OF SECTORS

The area of a sector is given as:
 
θ
2
A
πr

360

Determine the area of a piece of pizza
with a central angle of 48° and a radius
of 10inches
A 
48

360
π 10   41.89 in
2

2
Ans
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AREAS OF SEGMENTS


A segment of a circle is a figure formed
by an arc and the chord joining the end
points of the arc
The area of a segment is found by
subtracting the area of a triangle from
the area of a sector

That triangle is always isosceles

2 radii for sides
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AREAS OF SEGMENTS

Find the area of segment ACB in the
figure below, given that AOB = 85,
the radius of the circle is 4 in, and AB is
10 in:
– Area of the sector:
A = (85°/360°)(4")2 = 11.868 in2
C

A
B
O
2"
– Area of the triangle:
A = ½ bh = ½ (10")(2") = 10 in2
– Area of the segment:
A = 11.868 in2 – 10 in2 = 1.868 in2 Ans
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AREAS OF SEGMENTS

You build a nice deck in your rectangular back yard that
is circular right outside the sliding doors. The yard is 50
ft long and 100 ft wide. The sides by the patio are 15
and 30 feet. The angle for the patio is 135° and the
radius is 31.5 ft. Find the grass area in the yard.
Area of Yard
A  100 * 50
A  5000ft
2
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AREAS OF SEGMENTS

You build a nice deck in your rectangular back yard that is circular right outside the
sliding doors. The yard is 50 ft long and 100 ft wide. The sides by the patio are 15 and
30 feet. The angle for the patio is 135° and the radius is 31.5 ft. Find the grass area in
the yard.

So the yard is 5000 square feet without a deck so what does the
deck take out?
Area of Sector
A 
135
 31.5 2
360
A  1168.97ft
2
Area of Triangle
need a height
2
27.5  b
2
 31.5
2
b  15.36ft
1
A  55 15.36 
2
A  422.4ft
2
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AREAS OF SEGMENTS

You build a nice deck in your rectangular back yard that is circular right outside the
sliding doors. The yard is 50 ft long and 100 ft wide. The sides by the patio are 15 and
30 feet. The angle for the patio is 135° and the radius is 31.5 ft. Find the grass area in
the yard.

So the area of the sector minus the area of triangle will give us the
segment or the deck.
1168.97 – 422.4 = 746.57 square feet
5000 – 746.57 = 4256.43 square feet of grass remaining after the
desk is installed


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AREAS OF ELLIPSES



An ellipse is a closed oval-shaped curve that is
symmetrical to two lines or axes that are
perpendicular to each other
The longer axis is called the major axis and the
shorter axis is called the minor axis
The area of an ellipse is equal to the product of 
and one half the major axis and one half the minor
axis
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AREAS OF ELLIPSES

Find the surface area of an elliptical
dining table that is 8 feet long (major
axis) and 5 feet wide (minor axis):
Area = (8 ft  2)(5 ft  2)
= 31.416 ft2 Ans
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PRACTICE PROBLEMS
1.
2.
3.
4.
Find the area of a circle that has a radius of 7.25
meters.
Determine the radius of a circular patio which is to
have an area of 14 square yards.
The radii of two circles are 6 inches and 2 inches.
Compare the area of the larger circle with the area
of the smaller circle.
Find the area of the sector of a circle with a central
angle of 78° and a radius of 4.5 inches.
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PRACTICE PROBLEMS (Cont)
5.
6.
7.
8.
Determine the central angle for a sector of a circle
with a 3-meter radius given that the area of the
sector is 3.77 square meters.
Find the area of a segment of a circle given a central
angle of 60° and a radius of 4 inches when the height
and base of the triangular section are 2 inches and 3
inches respectively.
Determine the area of an ellipse with a major axis of
7.5 cm and a minor axis of 5.5 cm.
Determine the major axis of an ellipse if its area is
236.34 square yards and the minor axis is 12.3
yards.
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PRACTICE PROBLEMS (Cont)
9.
In a circular tank there is this cross
sectional view and measurements.
What is the area of the water in the
bottom of the tank?
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PROBLEM ANSWER KEY
1.
2.
3.
4.
5.
6.
7.
8.
9.
165.13 m2
2.111 yards
The area of the larger circle is 9 times
larger than the area of the smaller
circle
13.784 in2
48°
5.378 in2
32.4 cm2
24.465 yards
17.39 yards2
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