In order to solve problems which require application of the area
(circumference) for circles, it is necessary to
A typical problem involving the area
of a circle
gives us the area, circumference
and/or lengths of the radius
or diameter. We may also be given a relationship between the area
of other figures inscribed in the circle. We need to calculate some of these quantities given information about the others. Two examples of this type of problem follow:
- Suppose the circumference of a circle is 5p. What are the radius and area of this circle?
is represented by the formula C = 2p
r and given as C = 5p
2pr = 5 p
r = 5/2
Knowing that r = 5/2, we can find area
A = pr2 = p(5/2)2 = 25p/4
We rely on the formulas for area
and circumference. Sometimes we don’t know the radius
when starting the problem. The radius
are the key measurements in any circle.
- Suppose a circle has a circumference of 28p. A right triangle with an altitude of 4x and a base of 7x is inscribed in this circle as shown in the diagram below. What are the diameter of this circle and the area of the shaded region?
Always make sure that you have a diagram in situations like this where another figure is inscribed in a circle. In our diagram, observe that the base b
of the triangle
is also a diameter
of the circle. We will use the fact that the diameter
d = 7x to help solve the problem.
pd = 28 p
d = 28
Earlier we learned that the diameter
of this circle
could be represented as 7x since it is the hypotenuse
of an inscribed right triangle.
7x = 28
x = 4
= ½bh = ½(7x)(4x) = 14x2
Therefore the area
of the shaded region