def
Area of a circle is the area enclosed by a circle.
circlearea
circumferemetalk
Area of a circle
circleareaform
where r = radius of circle


There is a relationship between the
area of a circle and its circumference.
First, remember that the circumference of a circle is C = 2πr,
and the area of a rectangle is A = bh.

A circle can be divided into congruent sectors (pie slices). The sectors are then pulled out of the circle and arranged as shown in the middle diagram. The length across the top (over the curved arcs) is half of the circumference. When placed in these positions, the sectors roughly form a parallelogram. The larger the number of sectors that are cut, the less curvy the arcs will appear and the more the shape will resemble an actual parallelogram. As seen in the last diagram, the parallelogram can be changed into a rectangle by slicing half of the last sector and placing it to the far left. We now have our sectors forming a rectangle.ac2
The area of a rectangle is A = bh.
Thus, the area of the sectors that make up the rectangle is πrr = πr2
The area of a circle: A = πr2

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Area of sectors of a circle
(Sectors are like "pizza" pie slices of a circle.)
areaChalf
Semi-circle
(half of circle = half of area)
areaCh
areaCquart
Quarter-circle
(¼ of circle = ¼ of area)
areaCq
areaCarc
Any Sector
(fractional part of the area)
areaCs
n = degrees in central angle of sector

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To refresh your memory on expressing answers involving π, see Circumference lesson.

exs

bullet Find the area of a circle with a radius of 14 inches.
    Express answer rounded to the nearest hundredth.
circle1
A = πr2
A = πr2 = π(14)2 = 615.7521601 = 615.75
Area = 615.75 square inches


bullet A square with a side length of 4 feet, has a circle cut out of its interior. The radius of the circle is 1.5 feet. To the nearest foot, what is the area of the remaining portion of the square?
circlebluesquare
Find the area of the square and subtract the area of the circle.
Area of square = 42 = 16
Area of circle = π(1.5)2 = 7.068583471
16 - 7.068583471 = 8.931416529 = 9 ft.
Answer = 9 feet


bullet Given: two circles with radii of 3 and 6. What is the ratio of the area of the smaller circle to the area of the larger circle?
circleratio
Notice that the ratio of the radii is 1:2, but the ratio of the areas is 1:4. The ratio of the areas is the square of the ratio of the radii.

Find the area of each circle.
Since not rounding is mentioned, leave the answer in terms of π.
A
(small) = π • 32 = 9π
A
(large) = π • 62 = 36π
Answer: ratio = circleratioG


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