Notes: Area of Circles I. Circles
A circle is the locus of points in a plane that are a fixed distance from a point called the center of the circle. A circle is named by the symbol and its center. A has radius r = AB and diameter d = CD. The irrational number is defined as the ratio of the circumference C to the diameter d, or
Solving for C gives the formula C = d. Also d = 2r, so C = 2r. You can use the circumference of a circle to find its area. Divide the circle and rearrange the pieces to make a shape that resembles a parallelogram.
The base of the parallelogram is about half the circumference, or r, and the height is close to the radius r. So A r · r = r2.
The more pieces you divide the circle into, the more accurate the estimate will be.
Ex 1a: Find the area of K in terms of .
A = r2 Area of a circle.
A = (3)2 Divide the diameter by 2 to find the radius, 3.
A = 9 in2 Simplify. Ex 1b: Find the radius of J if the circumference is (65x + 14) m.
C = 2r Circumference of a circle
(65x + 14) = 2r Substitute (65x + 14) for C.
r = (32.5x + 7) m Divide both sides by 2. Ex 1c: Find the circumference of M if the area is 25 x2 ft2 Step 1 Use the given area to solve for r.
A = r2 Area of a circle 25x2 = r2 Substitute 25x2 for A.
25x2 = r2 Divide both sides by .
5x = r Take the square root of both sides.
Step 2 Use the value of r to find the circumference. C = 2r C = 2(5x) Substitute 5x for r.
C = 10x ft Simplify. Ex 1d: Find the area of A in terms of in which C = (4x – 6) m.
A = r2 Area of a circle.
Divide the diameter by 2 2 A = (2x – 3) m to find the radius, 2x – 3.
A = (4x2 – 12x + 9) m2 Simplify.
Helpful Hint The key gives the best possible approximation for on your calculator. Always wait until the last step to round. Fun news story: http://www.npr.org/blogs/money/2014/02/26/282132576/74- 476-reasons-you-should-always-get-the-bigger-pizza
Ex 2a: A pizza-making kit contains three circular baking stones with diameters 24 cm, 36 cm, and 48 cm. Find the area of each stone. Round to the nearest tenth.
24 cm diameter 36 cm diameter 48 cm diameter
A = (12)2 A = (18)2 A = (24)2 ≈ 452.4 cm2 ≈ 1017.9 cm2 ≈ 1809.6 cm2 Ex 2b: A drum kit contains three drums with diameters of 10 in., 12 in., and 14 in. Find the circumference of each drum.
10 in. diameter 12 in. diameter 14 in. diameter C = d C = d C = d C = (10) C = (12) C = (14)
C = 31.4 in. C = 37.7 in. C = 44.0 in.