Substitute the radii into the formula for the area of a circle and evaluate.
D
What do the values in the last row of the table tell you?
A
2r
B
Ď€/4
C
Radius (units)
2
3
4
2r
Area of Circle (units^2)
4Ď€
9Ď€
16Ď€
4Ď€ r^2
Length of 1 Side of the Square
4
6
8
4r
Area of Square (units^2)
16
36
64
16 r^2
Ratio (Area of circle/Area of square)
Ď€/4
Ď€/4
Ď€/4
Ď€/4
D
The ratio is π4.
Practice makes perfect
We are given the following figure composed of a circle and a square. We want to find the length of one side of the square.
We are told that the circle touches the square at the midpoints of the four sides. From this we know that the radius of the circle is exactly half the side of the square.
The side of the square is exactly twice the radius.
We are given two formulas. One is a formula for the area of a circle A_c. The other is a formula for the area of a square A_s in terms of the radius of the inscribed circle.
A_c &= π r^2
A_s &= 4r^2
We want to write the ratio of the area of the circle A_c to the area of the square A_s in the simplest form. To do so, we will write the ratio as a fraction, then cancel out any common factors. Let's do it!
In Part A of the exercise we found that the length of one side of a square is always twice the radius of the circle. Let's include this in our table.
Radius (units)
2
3
4
2r
Area of Circle (units^2)
Ď€(2)^2 or 4Ď€
Length of 1 Side of the Square
2( 2)=4
2( 3)=6
2( 4)=8
2( 2r)=4r
Area of Square (units^2)
4^2 or 16
Ratio (Area of circle/Area of square)
Let's now substitute the radii into the formula for the area of a circle and evaluate. We will also calculate the areas of the squares. It will be useful to know that the area of a square is its side length squared.
Radius (units)
2
3
4
2r
Area of Circle (units^2)
Ď€( 2)^2=4Ď€
Ď€( 3)^2=9Ď€
Ď€( 4)^2=16Ď€
Ď€( 2r)^2=4Ď€ r^2
Length of 1 Side of the Square
4
6
8
4r
Area of Square (units^2)
( 4)^2=16
( 6)^2=36
( 8)^2=64
( 4r)^2=16r^2
Ratio (Area of circle/Area of square)
At last, we can calculate the ratio between the areas, just like we did in Part B.
Radius (units)
2
3
4
2r
Area of Circle (units^2)
4Ď€
9Ď€
16Ď€
4Ď€ r^2
Length of 1 Side of the Square
4
6
8
4r
Area of Square (units^2)
16
36
64
16r^2
Ratio (Area of circle/Area of square)
4Ď€/16=Ď€/4
9Ď€/36=Ď€/4
16Ď€/64=Ď€/4
4Ď€ r^2/16 r^2=Ď€/4
We want to draw a conclusion about the relationship between the areas of the circle and the square. Let's take a look at the table from Part C.
Radius (units)
2
3
4
2r
Area of Circle (units^2)
4Ď€
9Ď€
16Ď€
4Ď€ r^2
Length of 1 Side of the Square
4
6
8
4r
Area of Square (units^2)
16
36
64
16 r^2
Ratio (Area of circle/Area of square)
Ď€/4
Ď€/4
Ď€/4
Ď€/4
We see that in all four cases the ratio between the areas is π4. We can then conclude that the ratio always equals π4.
