📐 Understanding Perimeter Ratio
The perimeter ratio of two similar polygons is the ratio of their corresponding side lengths. If two polygons are similar, it means they have the same shape but can be different sizes. The ratio of their perimeters is directly proportional to the ratio of their corresponding sides.
- 📏Definition: It's the comparison of the distances around two similar figures.
- 🧮Calculation: If polygon A has a perimeter of $P_A$ and polygon B has a perimeter of $P_B$, then the perimeter ratio is $\frac{P_A}{P_B}$. This ratio is equal to the scale factor between the two polygons.
- 🔗Relationship to Sides: If the ratio of corresponding sides of two similar polygons is $a:b$, then the ratio of their perimeters is also $a:b$.
📏 Understanding Area Ratio
The area ratio of two similar polygons is the ratio of their areas. While the perimeter ratio is a direct comparison of side lengths, the area ratio involves the square of the scale factor.
- 📐Definition: It's the comparison of the amount of surface covered by two similar figures.
- AreaCalculation: If polygon A has an area of $A_A$ and polygon B has an area of $A_B$, then the area ratio is $\frac{A_A}{A_B}$. This ratio is equal to the square of the scale factor between the two polygons.
- 📈Relationship to Sides: If the ratio of corresponding sides of two similar polygons is $a:b$, then the ratio of their areas is $a^2:b^2$.
📊 Perimeter Ratio vs. Area Ratio: A Comparison Table
| Feature |
Perimeter Ratio |
Area Ratio |
| Definition |
Ratio of the lengths of the perimeters of two similar polygons. |
Ratio of the areas of two similar polygons. |
| Relationship to Side Lengths |
If the ratio of corresponding side lengths is $a:b$, the perimeter ratio is $a:b$. |
If the ratio of corresponding side lengths is $a:b$, the area ratio is $a^2:b^2$. |
| Calculation |
$\frac{P_1}{P_2}$, where $P_1$ and $P_2$ are the perimeters of the polygons. |
$\frac{A_1}{A_2}$, where $A_1$ and $A_2$ are the areas of the polygons. |
| Scale Factor |
Equal to the scale factor between the polygons. |
Equal to the square of the scale factor between the polygons. |
🔑 Key Takeaways
- 💡Perimeter Ratio: Deals with the lengths around the polygon and is directly proportional to the side lengths.
- 📐Area Ratio: Deals with the space inside the polygon and is proportional to the square of the side lengths.
- ➕Relationship: If you know the scale factor (ratio of corresponding sides), you can easily find both the perimeter and area ratios. Perimeter ratio is the scale factor, and the area ratio is the scale factor squared.