A
circle is a planar figure with all points the same distance from a fixed point. That fixed point is called the
center of the circle. Any segment that goes from the center to a point on the circle is called a
radius of the circle. A
diameter is any segment that passes through the center and has its endpoints on the circle. Obviously, a diameter is twice as long as a radius. In Figure
12 ,
O is the center,
OB
,
OC
, and
OA
each a radius, and
AC
is a diameter.
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Figure 12
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A circle with center, radius, and diameter labeled.
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Finding the Circumference
In ancient times the Greeks discovered that for all circles, the circumference divided by the diameter always turns out to be the same constant value. The Greek letter π (pi) is now used to represent that value. In fractional or decimal form, the commonly used approximations of π are π ≈ 3.14 or π ≈ 22/7. The Greeks found the formula
Ccircle/d = π, which is rewritten in the following form.
If you briefly regard a circle as a regular polygon with infinitely many infinitesimally
small
sides, you see that the apothem and radius become the same length (Figure
13 ).
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Figure 13
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Apothem and radius of a circle.
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Finding the Area
Looking at the area formula for a regular polygon and making the appropriate changes with regard to the circle,
That is, the formula for the area of a circle now becomes the following:
Example 1: Find the circumference and area for the circle in Figure
14 . Use 3.14 as an approximation for π.
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Figure 14
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Finding the circumference and area of a circle.
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Example 2: If the area of a circle is 81π ft2, find its circumference.
So the circumference is approximately 56.52 ft.
Fundamental Ideas
Perimeter and Area
