Learning Math: Measurement
Circles and Pi (π) Part B: Area of a Circle (60 minutes)
Session 7, Part B
In This Part
- Transforming a Circle
- Examining the Formula
Transforming a Circle
Let’s further examine the formula for area of a circle, A = • r^{2}. How do we interpret the symbols r^{2}? If r is the radius of a circle, then r^{2} is a square with sides of length r. Examine the circles below. A portion of each circle is covered by a shaded square. We can call each of these squares a radius square.
Problem B5
Use the circles (PDF document) to work on this problem. For each circle, cut out several copies of the radius square from a separate sheet of centimeter grid paper. Determine the number of radius squares it takes to cover each circle. You may cut the radius squares into parts if you need to. Record your data in the table below.
Circle |
Radius of Circle |
Area of Radius Square |
Area of Circle |
Number of Radius Squares Needed |
1 | ||||
2 | ||||
3 |
Examining the Formula
Let’s further examine the formula for area of a circle, A = π • r^{2}. How do we interpret the symbols r^{2}? If r is the radius of a circle, then r^{2} is a square with sides of length r. Examine the circles below. A portion of each circle is covered by a shaded square. We can call each of these squares a radius square.
Problem B5
Use the circles (PDF document) to work on this problem. For each circle, cut out several copies of the radius square from a separate sheet of centimeter grid paper. Determine the number of radius squares it takes to cover each circle. You may cut the radius squares into parts if you need to. Record your data in the table below.
Circle |
Radius of Circle |
Area of Radius Square |
Area of Circle |
Number of Radius Squares Needed |
1 | ||||
2 | ||||
3 |
Problem B6
- What patterns do you observe in your data?
- If you were to estimate the area of any circle in radius squares, what would you report as the best estimate?
Problem B7
Does the activity of determining the number of radius squares it takes to cover a circle provide any insights into the formula for the area of a circle?
Problem B8
When you enlarge a circle so that the radius is twice as long (a scale factor of 2), what do you think happens to the circumference and the area? Do they double? Experiment by enlarging circles with different radii and analyzing the data.
Take It Further
Problem B9
Experiment by enlarging a circle by a scale factor of 3, by a scale factor of 2/3, and by a scale factor of k. Generalize your findings.
Take It Further
Problem B10
If a circle has a radius of 5 cm and the margin of error in measurement is 0.2 cm, what is a reasonable approximation for the area of the circle?
Solutions
Problem B1
The area of the figure is exactly the area of the circle, since no area has been removed or added, only rearranged.
Problem B2
The length of the base is one-half the circle’s circumference, since the entire circumference comprises the scalloped edges that run along the top and bottom of the figure, and exactly half of it appears on each side. The base length is C/2.
Problem B3
Since the circumference is 2 • π • r, where r is the radius, the base is half of this. The base length is π • r.
Problem B4
As the number of wedges increases, each wedge becomes a nearly vertical piece. The base length becomes closer and closer to a straight line of length π • r (or half the circumference), while the height is equal to r. The area of such a rectangle is π • r • r, or π • r^{2}.
Problem B5
Here is the completed table:
Circle |
Radius of Circle |
Area of Radius Square |
Area of Circle |
Number of Radius Squares Needed |
||||
1 | 6 | 36 | 36 • π | A little more than 3 | ||||
2 | 4 | 16 | 16 • π | A little more than 3 | ||||
3 | 3 | 9 | 9 • π | A little more than 3 |
Problem B6
- In each case, it takes a little more than three radius squares to form the circle. If using approximations, it should always take around 3.14 of the squares to cover the circle.
- The best estimate is somewhere between 3.1 and 3.2, which we know is roughly the value of π.
Problem B7
The formula for the area of a circle is A = π • r^{2}. The activity helps one understand that a bit more than three times a radius square is needed to cover the circle. Namely, it illustrates why the formula is π • r^{2}.
Problem B8
Think about a circle with a radius equal to 1 (r = 1). The circumference and the area of this circle are as follows:
C = 2 • 1 • π = 2π
A = 1^{2} • π = π
Now double the radius to 2 units (r = 2). The circumference and the area of the new circle are as follows:
C = 2 • 2 • π = 4π
A = 2^{2} • π = 4π
The circumference of the new circle doubled, but the area is multiplied by a factor of 4 (the square of the scale factor). You can replace the 1 with any other number, or with a variable r, to see that this relationship will always hold.
Problem B9
Scale factor 3:
C = 2π • (3r) = 6 πr
A = π • (3r)^{2} = 9 π r^{2}
Scale factor 2/3:
C = 2π • (2/3r) = (4/3)πr
A = π • (2/3r)^{2} = (4/9)πr^{2}
Scale factor k:
C = 2π • (kr) = k(2πr)
A = π • (kr)^{2} = k^{2π}r^{2}
As with other similar figures, the circumference (or perimeter) of the shape is multiplied by the scale factor, while the area is multiplied by the square of the scale factor. This is also evident in the formulas for each; the circumference formula involves r, while the area formula involves r^{2}.
Problem B10
A reasonable approximation is 25π cm, but the margin of error will be larger than 0.2 cm. The actual area in square centimeters may be anywhere from (4.8)^{2π} (lower limit) to (5.2)^{2π} (upper limit). Since 4.8^{2}, or 23.04, and 5.2^{2}, or 27.04, are each about 2 units away from 5^{2}, the margin of error for the area is approximately 2π cm^{2}.