MM2G3 Students will understand properties of circles.
Areas of Circles and
Sectors
Wednesday, March 23, 2016
How do we find the areas of
circles and sectors?
Lesson 6.8
M2 Unit 3: Day 10
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Daily Homework Quiz
Daily Homework Quiz
Find the indicated measure.
1.
Circumference
2.
Radius
C = 48 ft
ANSWER
about 81.68 in.
ANSWER
about 7.64 ft
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Daily Homework Quiz
Daily Homework Quiz
3.
Length of AB
4.
ANSWER
Find the total circumference of the circles.
8.64 cm
ANSWER
100.53 cm
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Daily Homework Quiz
Daily Homework Quiz
Find the indicated measure.
5.
6.
Radius
ANSWER
about 21.88 cm
Circumference
ANSWER
about 222.72 in.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE
Use the formula for area of a circle
Find the indicated measure.
SOLUTION
A = πr2
a.
Area
Write formula for the
area of a circle.
= π (2.5)2
Substitute 2.5 for r.
= 6.25π
Simplify.
≈ 19.63
Use a calculator.
r = 2.5 cm
ANSWER
The area about 19.63 square centimeters.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE 1
EXAMPLE
Use the formula for area of a circle
Find the indicated measure.
SOLUTION
b.
A = πr2
Write formula for the
area of a circle.
113.1 = πr2
Substitute 113.1 for A.
113.1
= r2
π
6
≈ r
Divide each side by π.
Diameter
Find the positive square root of
each side.
A = 113.1 cm2
ANSWER
The radius is about 6 centimeters, so the diameter is
about 12 centimeters.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Sector of a Circle:
the region bounded by two radii of
the circle and their intercepted arc
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Area of a Sector: a portion of
the area of the whole circle
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE 2
EXAMPLE
Find areas of sectors
Find the areas of the sectors formed by
UTV.
SOLUTION
STEP 1
Find the measures of the minor and major arcs.
Because m UTV = 70°, mUV = 70° and
mUSV = 360° – 70° = 290°.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE 2
EXAMPLE
Find areas of sectors
Find the areas of the sectors formed by
UTV.
STEP 2
Find the areas of the small and large sectors.
Area of small sector = mUV°
360
°
70
=
360°
≈ 39.10
πr2
π
Write formula for area
of a sector.
82
Substitute.
Use a calculator.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE 2
EXAMPLE
Find areas of sectors
2
Area of large sector = mUSV
πr
360°
°
290
=
π
°
360
ANSWER
≈ 161.97
Write formula for area
of a sector.
82
Substitute.
Use a calculator.
The areas of the small and large sectors are about
39.10 square units and 161.97 square units,
respectively.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED
PRACTICE
Guided Practice
Use the diagram to find the indicated measure.
1.
Area of
D
SOLUTION
A = πr2
Write formula for the area of a circle.
= π (14)2
Substitute 14 for r.
= 196π
Simplify.
= 617.75
Use a calculator
The area of
D is about 615.75 ft2.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED
PRACTICE
Guided Practice
Use the diagram to find the indicated measure.
2.
Area of red sector
SOLUTION
STEP 1
Find the measures of major arcs.
Because m FDE = 120, mFE = 120 and mFGE
= 360 – 120 = 140.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED
PRACTICE
Guided Practice
STEP 2
Find the area of red sector.
Area of red sector = mFE π r2
360
=
120
360
= 205.25
π
Write formula for area
of a sector.
142 Substitute.
Use a calculator.
ANSWER
The area of red sector is about 205.25 ft2.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED
PRACTICE
Guided Practice
Use the diagram to find the indicated measure.
3.
Area of blue sector
SOLUTION
STEP 1
Find the measure of the blue arc.
Because m FDE = 120, MFE = 120 and mFGE
= 360 – 120 = 240.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Guided Practice
STEP 2
Find the area of blue sector.
Area of blue sector = mFEG π r2 Write formula for area
360
of a sector.
= 240
360
π
= 410.50 ft2
142
Substitute.
Use a calculator.
ANSWER
Area of blue sector is about 410.50 ft2.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE
Use the Area of a Sector Theorem
Use the diagram to find the area of
V.
SOLUTION
Area of sector TVU = mTU
360°
°
40
35 =
360°
315 = Area of
Area of V Write formula for
area of a sector.
Area of V Substitute.
V
Solve for Area of
ANSWER
The area of
V is 315 square meters.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
V.
MM2G3 Students will understand properties of circles.
EXAMPLE 4
Standardized Test Practice
SOLUTION
The area you need to paint is the area of the
rectangle minus the area of the entrance. The
entrance can be divided into a semicircle and a
square.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
EXAMPLE 4
Standardized Test Practice
=
°
180
36(26) –
360°
(π
82 ) + 162
= 936 – [32π + 256]
≈ 579.47
The area is about 579 square feet.
ANSWER
The correct answer is C.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED
PRACTICE
Guided Practice
4.
Find the area of
H.
SOLUTION
Area of sector FHG =
mFG
360
Area of
H Write formula
for area of a
sector.
85
Area of H Substitute.
360
907.92 = Area of H
Solve for Area
214.37 =
of
H.
ANSWER
The area of
H is 907.92 cm2.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED
PRACTICE
Guided Practice
5.
Find the area of the figure.
SOLUTION
STEP 1
1
A = 2
1
= 2
Take the top as base, which is 7 m
and find the area of the triangle
b
h
7 7
Write formula for area of a
triangle.
Substitute.
= 24.5
= 24.5 m
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED
PRACTICE
Guided Practice
STEP 2
find the area of the semicircle
1
A=
π r2
2
1
=
π (3.5)2
2
= 19.5
STEP 3
Write formula for area of a
sector.
Substitute.
multiply
Add the areas
Area of figure = Area of triangle + Area of semicircle
= 24.5 + 19.5
= 43.74 m2
ANSWER
The area of the figure is about 43.74 m2.
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
GUIDED
PRACTICE
Guided Practice
6.
If you know the area and radius of a sector of a
circle, can you find the measure of the
intercepted arc? Explain.
SOLUTION
yes; the formula for the area of sector is
m
A = 360 π r2 and if you solve this
for m, you get 360A
π r2
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
MM2G3 Students will understand properties of circles.
Homework:
Page 232 (#1-15 all)
MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.