Math Background

Pi: Overview

There are many fascinating numbers in mathematics. One of the most interesting number relationships that students can discover in geometry is that the ratio of the circumference of a circle (the distance around a circle) to the diameter of a circle (the length of a line across a circle that passes through its center) is approximately equal to 3.14 or twenty-two-sevenths. This ratio is the same for all circles. It is called pi, and is represented by the Greek letter pi.

It is important that your students feel comfortable with the concept of pi by actually measuring the circumference and diameter of different circles with string and rulers. Use any circular objects in your classroom, such as clocks, coins, CDs, and so on. Students can create a table like the one below to verify for themselves that this relationship is valid.

table

Students should notice that as in all measurements, the values are not exact. So it is highly unlikely that pi will come out to exactly 3.14 for any of the circles they measured. However, the average of the ratios might come out quite close to 3.14.

Your students may be interested in learning more facts about pi. Many mathematicians celebrate Pi Day each year on March 14 (3/14). March 14 is also Albert Einstein's birthday! Two mathematicians, Yasumasa Kanada and Daisuke Takahashi from the University of Tokyo, calculated pi to 206,158,430,000 decimal places in 1999.

Since the ratio of the circumference to the diameter in every circle equals pi, it is easy to determine the formula for finding circumference, C = pi d. Students should remember that a diameter is twice the measure of a radius, so the formula C = 2pi r can also be used. One of the most widely used skills in algebra is to substitute values for variables in a formula and solve for the unknown variable. So having students find the circumference, given the value of the diameter or radius, helps prepare them for algebra.

The formula for the area of a circle is a little more complex than the formula for circumference. Look at the circle below. It has been cut into 8 parts and rearranged in a shape that resembles a parallelogram.

circle cut into eight parts and rearranged

We can use the formula for the area of a parallelogram, A = b x h, to develop the formula for the area of a circle. The base of the parallelogram above equals half the circumference of the circle, or b = C over two. Since C = 2pi r,we can substitute 2pi r for C to show that b = pi r. The height of this parallelogram is a radius of the circle, or r, so the area of the circle would be A = pi r x r, or pi r².


Houghton Mifflin Math Grade 6