Rewrite the ratio of the circumferences using the circumference formula.
E
e
Remember that in similar figures the ratio of the corresponding lengths is equal to the scale factor.
F
f
Use the equation you found in the previous part.
A
a
Example Solution:
B
b
Example Solution:
Radius
Circumference
0.7
≈ 4.40
1.4
≈ 8.80
2.8
≈ 17.59
C
c
All circles are similar to each other.
D
d
The ratio of the circumferences of circles is also 2.
E
e
C_B=b/aC_A
F
f
4 inches
Practice makes perfect
a
We are asked to draw three circles in which the scale factor from each circle to the next larger circle is 1:2. First, let's draw a point C that will be the center of all three circles and a point X that lies on the line l. These two objects will help us find the next radii.
Now let's draw the first circle centered at C using a compass.
Next, keeping the same compass setting we will put the compass at X and draw an arc on line l. Then we will repeat the process, but this time putting the compass at the point of intersection of the arc and the line.
Let's call the point of intersection of the second arc and the line Y. Since XY has a length that is the doubled radius of the first circle, this segment will be the radius of our second circle.
To draw our second circle we will copy XY using the compass, then put the compass at C and draw a circle.
Now, keeping the same compass setting we will draw an arc on our line starting at Y. We will name the point of intersection of this arc and the line Z.
Notice that XZ=2XY, which means that XZ will be the radius of the third circle. Let's copy XZ, put the compass at C, and draw the last circle.
This is one example of three circles with the desired scale factor.
b
In this part we are asked to calculate the radius and the circumference of each circle. First let's draw the radius of each circle.
Next, using the ruler we will measure the length of each radii.
Next let's recall that the circumference of a circle is the doubled product of the radius of a circle and pi. Using this information, we will evaluate the circumference of each circle and record our results in a table.
Radius
2Ï€ r
Circumference
0.7
2Ï€ ( 0.7)
≈ 4.40
1.4
2Ï€ ( 1.4)
≈ 8.80
2.8
2Ï€ ( 2.8)
≈ 17.59
c
Let's begin with recalling that all circles are geometrically similar by the Similar Circles Theorem. Since our three figures are circular they are similar to each other.
d
In this part we want to make a conjecture about the ratio between the circumferences of two circles when the ratio between the radii is 2. To do this, let's rewrite this ratio using the circumference formula.
C_1/C_2=2Ï€ r_1/2Ï€ r_2=r_1/r_2= 2
As we can see, the ratio of the circumferences of the circles is equal to the ratio between the radii of these circles.
e
Now we are given that the scale factor from ∘ A to ∘ B is ba and we want to write an equation relating the circumference C_A to the circumference C_B. Let's recall that in similar figures the ratio of the corresponding lengths is equal to the scale factor.
C_B/C_A= b/a ⇒ C_B=b/aC_A
f
In this part we want to find the circumference of ∘ B knowing that the scale factor from ∘ A to ∘ B is 13 and the circumference of ∘ A is 12 inches. To do this we will use the equation we found in the previous part. Let's substitute the values we know and solve for C_B.
