o Identify some basic parts of a circle, such as the radius and diameter
o Calculate the circumference of a circle
o Calculate the area of a circle
In this article, we will consider a geometric figure that does not involve line segments, but is instead curved: the circle. We will apply what we know about algebra to the study of circles and thereby determine some of the properties of these figures.
Introduction to Circles
Imagine a point P having a specific location; next, imagine all the possible points that are some fixed distance r from point P. A few of these points are illustrated below. If we were to draw all of the (infinite number of) points that are a distance r from P, we would end up with a circle, which is shown below as a solid line.
Thus, a circle is simply the set of all points equidistant (that is, all the same distance) from a center point (P in the example above). The distance r from the center of the circle to the circle itself is called the radius; twice the radius (2r) is called the diameter. The radius and diameter are illustrated below.
The Circumference of a Circle
As with triangles and rectangles, we can attempt to derive formulas for the area and "perimeter" of a circle. Unlike triangles, rectangles, and other such figures, the distance around the outside of the circle is called the circumference rather than the perimeter-the concept, however, is essentially the same. Calculating the circumference of a circle is not as easy as calculating the perimeter of a rectangle or triangle, however. Given an object in real life having the shape of a circle, one approach might be to wrap a string exactly once around the object and then straighten the string and measure its length. Such a process is illustrated below.
Obviously, as we increase the diameter (or radius) of a circle, the circle gets bigger, and hence, the circumference of the circle also gets bigger. We are led to think that there is therefore some relationship between the circumference and the diameter. As it turns out, if we measure the circumference and the diameter of any circle, we always find that the circumference is slightly more than three times the diameter. The two example circles below illustrate this point, where D is the diameter and C the circumference of each circle.
Again, in each case, the circumference is slightly more than three times the diameter of the circle. If we divide the circumference of any circle by its diameter, we end up with a constant number. This constant, which we label with the Greek symbol π (pi), is approximately 3.141593. The exact value of π is unknown, and it is suspected that pi is an irrational number (a non-repeating decimal, which therefore cannot be expressed as a fraction with an integer numerator and integer denominator). Let's write out the relationship mentioned above: the quotient of the circumference (C) divided by the diameter (D) is the constant number π.
We can derive an expression for the circumference in terms of the diameter by multiplying both sides of the expression above by D, thereby isolating C.
Because the diameter is twice the radius (in other words, D = 2r), we can substitute 2r for D in the above expression.
Thus, we can calculate the circumference of a circle if we know the circle's radius (or, consequently, its diameter). For most calculations that require a decimal answer, estimating π as 3.14 is often sufficient. For instance, if a circle has a radius of 3 meters, then its circumference C is the following.
The answer above is exact (even though it is written in terms of the symbol π). If we need an approximate numerical answer, we can estimate π as 3.14. Then,
Interested in learning more? Why not take an online class in Pre-Algebra?
The symbol ≈ simply means "approximately equal to."
Practice Problem: A circle has a radius of 15 inches. What is its circumference?
Solution: Let's start by drawing a diagram of the situation. This approach can be very helpful, especially in situations involving circles, where the radius and diameter can easily be confused.
Step-by-step explanation:
