Area and Perimeter #3 (Circles) - LS Home Page
Area and Perimeter #3 (Circles) - LS Home Page
Area and Perimeter #3 (Circles) - LS Home Page
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<strong>Circles</strong> page #1<br />
Circumference of a Circle<br />
The distance around a circle is its circumference.<br />
C = πd or C = 2πr where π = 3.14<br />
Example 1:<br />
1. The approximate circumference:<br />
7 cm<br />
C = 3.14(14) = 43.96 cm.<br />
2. The exact circumference:<br />
C = 14 π<br />
Example 2: A flagpole has a circumference of 30 cm. Find its diameter.<br />
Use 3.14 for π.<br />
C = πd<br />
30 = 3.14d<br />
30<br />
3.14 = 3.14d<br />
3.14<br />
d = 9.6 cm.<br />
Example 3: The sides of a square are 10 cm. Find the circumference of the<br />
circumscribed circle.<br />
10 cm<br />
d<br />
10 cm<br />
d 2 = 10 2 +10 2<br />
or<br />
d 2 = 100 + 100<br />
d 2 = 200<br />
n 2<br />
d = 200<br />
d =10 2 d =10 2<br />
n<br />
n<br />
Therefore, C = π10 2 = 10π 2
Problems:<br />
<strong>Circles</strong> page #2<br />
1. Find the exact circumference for each circle:<br />
A. B.<br />
5 cm<br />
8 cm<br />
2. To refinish a round table, a narrow strip of material is to be glued around the edge.<br />
The table has a 42 inch diameter. How long must the strip be?<br />
3. The rectangle is inscribed in the circle. Find the circumference of the circle.<br />
30 cm<br />
40 cm<br />
4. What is the perimeter of an enclosed semicircular area where the radius of the<br />
corresponding circle is 5? (Draw a sketch.)
<strong>Circles</strong> page <strong>#3</strong><br />
Example 4: Find the circumference of the circle whose center is (-2,1) <strong>and</strong> (2,4) is a<br />
point on the circle.<br />
r<br />
4<br />
3<br />
We need to find the radius with the pythagorean<br />
theorem or the distance formula.<br />
r 2 = 3 2 + 4 2<br />
r 2 = 9 +16<br />
r 2 = 25<br />
r = 5<br />
r =<br />
( 2 −−2) 2 + ( 4 −1) 2<br />
r = ( 4 ) 2 + ( 3) 2<br />
r = 16 +9<br />
r = 25 = 5<br />
The diameter is 2(5) = 10.<br />
The circumference is 10 π.<br />
Example 5: Find the circumference of the circle whose equation is (x-1) 2 +(y+2) 2 =16.<br />
Remember: Equation of the circle: (x − h) 2 + (y − k) 2 = r 2<br />
The center of our circle is (1,-2) <strong>and</strong> the radius is 4.<br />
The diameter is 8.<br />
The circumference is 8 π.<br />
center: (h,k) radius = r<br />
Problems:<br />
1. Find the circumference of the circle whose center is (3,2) <strong>and</strong> (5,1) is a point on the<br />
circle.<br />
2. Find the circumference of the circle whose equation is (x-3) 2 +(y+5) 2 =36.
3. A gardener is fencing both sides of a circular walk around the garden shown. How<br />
much fence does he need?<br />
Walk<br />
<strong>Circles</strong> page #4<br />
Garden<br />
3.5 feet<br />
49 feet<br />
Garden<br />
4. Joe started jogging around a circular track. Halfway around he got tired <strong>and</strong> walked<br />
back to his starting point. If the diameter of the track is 45 yards, how far did he go?<br />
45 yards<br />
start<br />
5. The sides of a square are 6 cm. Find the circumference of the circumscribed circle.<br />
6. The rectangle is inscribed in the circle. Find the circumference of the circle.<br />
30°<br />
2 cm
<strong>Circles</strong> page #5<br />
<strong>Area</strong> of a Circle: A=πr 2<br />
Example 1: Find the area of a circle with a radius of 6 cm.<br />
A = πr 2 = π6 2 = 36 π cm 2<br />
Example 2: The area of a circle is 70,650 in 2 . Find the radius to the nearest tenth of<br />
an inch. Use 3.14 for π.<br />
A = πr 2<br />
70, 650 = 3.14r 2<br />
70, 650<br />
3.14 = 3.14r2<br />
3.14<br />
22,500 = r 2<br />
r =<br />
22,500 =150 inches<br />
Example 3: Find the area of the shaded region:<br />
4 cm<br />
4 cm<br />
3 cm<br />
d 3 cm<br />
d 2 = 3 2 + 4 2<br />
d 2 = 9 + 16<br />
d 2 = 25<br />
d = 5<br />
radius = 2.5 cm<br />
Shaded <strong>Area</strong> = <strong>Area</strong> of the circle - <strong>Area</strong> of the rectangle<br />
Shaded <strong>Area</strong> = (3.14)(2.5) 2 - (3)(4)<br />
Shaded <strong>Area</strong> = 19.625 - 12<br />
Shaded <strong>Area</strong> = 7.625 cm 2<br />
Example 4: (x+3) 2 + (y-6) 2 = 16<br />
A. Find the center of the circle. (-3,+6)<br />
B. Find the radius of the circle. r = 4<br />
C. Find the circumference of the circle. C = πd C = 8 π<br />
D. Find the area of the circle. A = πr 2 A = 16 π
Problems:<br />
<strong>Circles</strong> page #6<br />
1. Find the area of a circle with a diameter of 6 cm.<br />
2. The area of a circle is 1962.50 in 2 . Find the radius to the nearest tenth of an inch.<br />
Use 3.14 for π.<br />
3. Find the area of the shaded region:<br />
8 cm<br />
6 cm<br />
4. (x+1) 2 + (y-8) 2 = 9<br />
A. Find the center of the circle.<br />
B. Find the radius of the circle.<br />
C. Find the circumference of the circle.<br />
D. Find the area of the circle.<br />
5. Given the area of a circle, find the radius <strong>and</strong> diameter. Use 3.14 for π.<br />
A = 380 cm 2<br />
radius: _________<br />
diameter: ________
<strong>Circles</strong> page #7<br />
6. Find the shaded area to the nearest tenth of a centimeter:<br />
A.The centers are O <strong>and</strong> P <strong>and</strong> the radius of the smaller circle is 4 cm. Use 3.14 for π.<br />
O<br />
P<br />
B.<br />
36 cm<br />
25 cm<br />
C. The square has a side of 4 inches <strong>and</strong> each circle has a radius of 1 inch.
7. (-1,-3) <strong>and</strong> (7,3) are the endpoints of a diameter of circle O.<br />
A. Draw the circle:<br />
<strong>Circles</strong> page #8<br />
Find:<br />
B. The center of the circle.<br />
C. The radius of the circle.<br />
D. The equation of the circle.<br />
E. The circumference of the circle.<br />
F. The area of the circle.<br />
8. Given the area of the circle, find the circumference.<br />
A = 16 π ft 2<br />
Circumference = _______________<br />
9. Given the circumference of the circle, find the area.<br />
C = 12 π cm<br />
<strong>Area</strong> = ________________<br />
10. A pipe has a circumference of 78.5 cm. Find its diameter. Use 3.14 for π.