CIRCLES

HINT: Pi =3.14

Geometry Lessons 8.5 and 8.6 Area of Circles, Sectors and Segments HW: 8.5/1-8, 12-14 8.6/1-12

Circumference   3in   C ≈ 18.8in

Finding a formulae for the area of a circle

    πr r

Area of Rectangle= Base x Height  

Finding the area given the diameter The radius of a circle is half of its diameter, or r = d 2 We can substitute this into the formula A = πr2

How to find the area of a circle given the radius Find the area of the circle below: Substitute the values for p 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 p 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 10 m A = πr2 A = πr2 = 3.14 × 22 = 3.14 × 52 Explain that rather than use the formula on the previous slide, it is usually easier to halve the diameter mentally to give the radius, before substituting it into the formula. The most common error is to neglect to half the diameter to find the radius and to substitute this value into the formula. Ensure that pupils do not make this mistake. = 12.56 cm2 = 78.5 m2 23 mm 78 cm A = πr2 A = πr2 = 3.14 × 232 = 3.14 × 392 = 1661.06 mm2 = 4775.94 cm2

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. Compare this with slide 74, which finds the perimeter of the same shape. Area of circle = 3.14 × 6.52 13 cm 6 cm = 132.665 cm2 Area of rectangle = 6 × 13 = 78 cm2 Total area = 132.665 + 78 = 210.665 cm2

The circumference of a circle is 18 The circumference of a circle is 18. Find the area of the circle, leave answer in terms of . C = 2  r A = r2 A = 92 A = 81 18 = 2r 9 = r

Arc Length Reminder 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.

m 2πr 360 . Arc Length Formula measure of the central angle or arc The circumference of the entire circle! 2πr Arc Length = 360 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 A region bounded by 2 radii and their intercepted arc. .

Area of a Sector Formula measure of the central angle or arc m˚ πr2 Area of a sector = 360 The area of the entire circle! The fraction of the 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 m2  164.54 m2

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 ft2  5.45 ft2

The area of sector AOB is and Find the radius of ○O. m Area of a sector = πr2 360 9 40 π = πr2 4 360 9 9 1 9 = r2 1 4 9 1 81 = r2 4 9 r = 2

The area of sector AOB is 48π and Find the radius of ○O. m πr2 Area of a sector = 360 270 48π = πr2 360 A O 4 16 3 4 r2 48 = 3 4 3 64 = r2 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 in2  80.42 in2

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 in2

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

 

  = ¼100π= 25π 10 =½∙10∙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 m2 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 = 4m. Simplify. = 8 m2

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 m2

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 in2  23.56 in2 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 m2

Find the area of the shaded region Find the area of the shaded region. Point O marks the center of the circle. 8. 9. 10. 11. 60˚ 8 12 60 O 6 4 30 O O 160 π units2 9π - 18 units2 24π - 36√3 units2 8π - 8√3 units2 3

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