HINT Pi 3 14 Geometry Lessons 5 7

HINT: Pi =3. 14

Geometry Lessons 5 -7 and 5 -8 Area of Circles, Sectors and Segments

Circumference 3 in C ≈ 18. 8 in

Finding a formulae for the area of a circle

Area of Rectangle= Base x Height

Finding the area given the diameter The radius of a circle is half of its diameter, or d r= 2 We can substitute this into the formula A = πr 2

How to find the area of a circle given the radius Find the area of the circle below: Substitute the values for and r.

How to find the area of a circle given the diameter Given the diameter: Radius = diameter divided by 2 Substitute the values for and r.

How would you find the shaded area? Find the area of the square and subtract the area of the circle.

The area of a circle Use π = 3. 14 to find the area of the following circles: 2 cm A = πr 2 = 3. 14 × A = πr 2 22 10 m = 12. 56 cm 2 A = πr 2 23 mm = 3. 14 × 52 = 78. 5 m 2 78 cm A = πr 2 = 3. 14 × 232 = 3. 14 × 392 = 1661. 06 mm 2 = 4775. 94 cm 2

Find the area of this shape Use π = 3. 14 to find area of this shape. The area of this shape is made up of the area of a circle of diameter 13 cm and the area of a rectangle of width 6 cm and length 13 cm. 6 cm 13 cm Area of circle = 3. 14 × 6. 52 = 132. 665 cm 2 Area of rectangle = 6 × 13 = 78 cm 2 Total area = 132. 665 + 78 = 210. 665 cm 2

The circumference of a circle is 18. Find the area of the circle, leave answer in terms of . C=2 r A = r 2 18 = 2 r A = 92 9=r A = 81

Arc Length Reminder n The length of part of the circumference. The length of the arc depends on what two things? 1) The measure of the arc. 2) The size of the circle. An arc length measures distance while the measure of an arc is in degrees.

Arc Length Formula measure of the central angle or arc m 2πr 360 The circumference of the entire circle! Arc Length = The fraction of the circle! .

REVIEW: Finding Arc Length Find the arc length. Give answers in terms of and rounded to the nearest hundredth. FG Use formula for area of sector. Substitute 8 for r and 134 for m. Simplify. 5. 96 cm 18. 71 cm

Sectors Let’s talk pizza

Sector of a circle n A region bounded by 2 radii and their intercepted arc. .

Area of a Sector Formula measure of the central angle or arc Area of a sector = m˚ 360 2 πr The fraction of the circle! The area of the entire circle! .

Finding the Area of a Sector Find the area of the sector. Give answers in terms of and rounded to the nearest hundredth. sector HGJ Use formula for area of sector. Substitute 12 for r and 131 for m. Simplify. = 52. 4 m 2 164. 54 m 2

Finding the Area of a Sector Find the area of the sector. Give answers in terms of and rounded to the nearest hundredth. sector ABC Use formula for area of sector. Substitute 5 for r and 25 for m. Simplify. 1. 74 ft 2 5. 45 ft 2

The area of sector AOB is Find the radius of ○O. m Area of a sector = πr 2 360 9 40 π= πr 2 4 360 99 1 2 9 = r 1 4 9 1 81 2 = r 4 r= 9 2 and

The area of sector AOB is 48π and Find the radius of ○O. m πr 2 Area of a sector = 360 270 48π = πr 2 360 4 16 3 2 4 48 = r 3 4 3 A O 64 = r 2 r =8 B

Finding the Area of a Sector Find the area of each sector. Give your answer in terms of and rounded to the nearest hundredth. sector JKL Use formula for area of sector. Substitute 16 for r and 36 for m. Simplify. = 25. 6 in 2 80. 42 in 2

Automobile Application A windshield wiper blade is 18 inches long. To the nearest square inch, what is the area covered by the blade as it rotates through an angle of 122°? Use formula for area of sector. r = 18 in. m˚ = 122˚ Simplify. 345 in 2

AREA OF SEGMENT A segment of a circle is a region bounded by an arc and its chord.

= ¼ 100π= 25π 10 =½∙ 10=50 Area of Segment = 25π - 50

Check It Out! Find the area of segment RST to the nearest hundredth. Step 1 Find the area of sector RST. Use formula for area of sector. Substitute 4 for r and 90 for m. = 4 m 2 Simplify.

Check It Out! Continued Find the area of segment RST to the nearest hundredth. Step 2 Find the area of ∆RST. ST = 4 m, and RS = 4 m. Simplify. = 8 m 2

Check It Out! Continued Find the area of segment RST to the nearest hundredth. Step 3 area of segment = area of sector RST – area of ∆RST = 4 – 8 4. 57 m 2

Lesson Quiz: Part I Find each measure. Give answers in terms of and rounded to the nearest hundredth. 1. area of sector LQM 7. 5 in 2 23. 56 in 2 2. length of NP 2. 5 in. 7. 85 in.

Lesson Quiz: Part II 3. The gear of a grandfather clock has a radius of 3 in. To the nearest tenth of an inch, what distance does the gear cover when it rotates through an angle of 88°? 4. 6 in. 4. Find the area of segment GHJ to the nearest hundredth. 55. 94 m 2

Find the area of the shaded region. Point O marks the center of the circle. 8. 9. 11. 10. 60˚ 8 160 3 π units 2 12 O 6 9π - 18 units 2 O 30 60 24π - 36√ 3 units 2 O 4 8π - 8√ 3 units 2

Some common fractions and measures! Arc or Central Angle Measure Fraction of the Circle 36 o 1/10 108 o 3/10 60 o 1/6 300 o 5/6 120 o 1/3 240 o 2/3 30 o 1/12 330 o 11/12 45 o 1/8 225 o 5/8