To find the scale factor of a perimeter, you simply divide the perimeter of the scaled shape by the perimeter of the original shape.
This method is directly related to the property of scaling geometric shapes:
- As stated in the provided reference, the perimeter of a geometric shape is calculated by multiplying the scale factor by the perimeter of the original perimeter.
In other words, the relationship is:
Original Perimeter × Scale Factor = New Perimeter
To isolate the scale factor and find its value, you rearrange this formula through division:
Scale Factor = New Perimeter / Original Perimeter
Understanding the Relationship
When a shape is scaled by a certain factor, all its linear dimensions, including its sides and perimeter, are multiplied by that same scale factor.
- If the scale factor is greater than 1, the shape (and its perimeter) gets larger.
- If the scale factor is between 0 and 1, the shape (and its perimeter) gets smaller.
- If the scale factor is exactly 1, the shape remains the same size.
For example, if a shape's perimeter is 10 units and it's scaled by a factor of 3, the new perimeter will be 10 × 3 = 30 units. Conversely, to find the scale factor if you know the original perimeter is 10 and the new perimeter is 30, you calculate 30 / 10 = 3.
Steps to Calculate the Scale Factor
Finding the scale factor of a perimeter is straightforward:
- Identify the Original Perimeter: Determine the perimeter of the initial, unscaled shape.
- Identify the New Perimeter: Determine the perimeter of the scaled shape.
- Divide: Divide the new perimeter by the original perimeter.
The result is the scale factor.
Example Calculation
Let's illustrate with an example:
Imagine you have a rectangular garden (the original shape) and you create a scaled-down model of it (the new shape).
- Original Garden Perimeter: 50 meters
- Scaled Model Perimeter: 10 meters
To find the scale factor used for the perimeter:
Scale Factor = New Perimeter / Original Perimeter
Scale Factor = 10 meters / 50 meters
Scale Factor = 0.2
In this case, the scale factor is 0.2 (or 1/5), indicating the scaled model's perimeter is 0.2 times the size of the original garden's perimeter. This also means the model is 1/5th the size of the garden in its linear dimensions.
Here's a quick look:
| Measurement | Original Shape | Scaled Shape |
|---|---|---|
| Perimeter (P) | 50 m | 10 m |
| Scale Factor (k) | - | P_new / P_original = 10 / 50 = 0.2 |
This demonstrates how dividing the new perimeter by the original perimeter reveals the scaling factor applied.
