How do you find the scale factor of two similar circles?
The Correct Answer and Explanation is :
To find the scale factor of two similar circles, you need to compare their corresponding measurements. In the case of circles, the primary corresponding measurement is the ratio of their radii (or diameters). Here’s how you can calculate the scale factor:
Step 1: Identify the radii or diameters of the two circles.
- Let’s say Circle 1 has a radius ( r_1 ) and Circle 2 has a radius ( r_2 ).
- Alternatively, you could use the diameters, where the diameter is twice the radius (( d = 2r )).
Step 2: Compute the ratio of the corresponding radii (or diameters).
- The scale factor is simply the ratio of the radius of Circle 2 to the radius of Circle 1.
[
\text{Scale factor} = \frac{r_2}{r_1}
]
If you are given the diameters, use the same formula:
[
\text{Scale factor} = \frac{d_2}{d_1}
]
Step 3: Interpret the scale factor.
- If the scale factor is greater than 1, Circle 2 is larger than Circle 1 by that factor.
- If the scale factor is less than 1, Circle 2 is smaller than Circle 1.
- If the scale factor is exactly 1, both circles are of the same size.
Example:
Suppose Circle 1 has a radius of 5 cm and Circle 2 has a radius of 10 cm.
The scale factor would be:
[
\text{Scale factor} = \frac{10}{5} = 2
]
This means that Circle 2 is 2 times larger than Circle 1.
Area and Scale Factor:
The area of a circle is related to its radius by the formula ( A = \pi r^2 ). If the scale factor of the radii is ( k ), the areas of the circles will be related by ( k^2 ). For example, if the radius of Circle 2 is 2 times that of Circle 1, then the area of Circle 2 will be 4 times the area of Circle 1.
In summary, the scale factor for similar circles is found by comparing their radii or diameters, and the ratio directly reflects the proportional difference in size.