208 Part 3: The Shape of the World
Area and Circumference
When you explored polygons, whether triangles, quadrilaterals, or polygons with more sides,
you always wanted to know two things: how to quickly find the distance around the edges of the
polygon (its perimeter), and how to measure the area enclosed by the polygon. You want to be
able to do these same things for the circle.
The circumference of a circle is the distance around the circle. The circumference is the circle’s
equivalent of the perimeter of a polygon. The distance around a circle is a little more than three
times the diameter of the circle. The formula for the circumference of a circle is C = Sd, where d
is the diameter of the circle and S (pi) is a constant approximately equal to 3.14159.DEFINITION
The circumference of a circle is the distance around the circle. The word
circumference comes from Latin. Circum is the preposition for around, and ferre is
a verb that means carry. If you carry something around the circle, you trace out the
circumference.You sometimes see the formula as C = 2Sr, because r is the radius of the circle, and the diameter
is twice the radius. Because S is an irrational number, many times you’ll give your answer
as a number times S, for example, 9S. If you use an approximate value, you’ll want to say
“approximately equal” when you give your answer. For most questions, you can use 3.14 or^22
7
as approximate values of S.
Suppose the circumference of circle O is 40S cm. To find its radius, use the circumference
formula Cd 40. The diameter must be 40 cm, and therefore the radius is half of that, or
20 cm.
The area of a circle, the space enclosed by the circle, can be approximated in a number of
ways. One is to sandwich it between a polygon whose sides are tangent to the circle, called a
circumscribed polygon, and a polygon whose vertices are on the circle, called an inscribed polygon.
The area of the circle will be larger than the area of the inscribed polygon but smaller than the
area of the circumscribed polygon. To get a good approximation, however, you would have to
use polygons with many sides and that makes for difficult calculations.DEFINITION
A polygon is circumscribed about a circle if the polygon surrounds the circle with
each side tangent to the circle. A polygon is inscribed in a circle if each of its vertices
lies on the circle.