๐ Understanding Scale Factors and Area
A scale factor is the ratio between corresponding measurements of an object and a representation of that object. When dealing with area, the effect of the scale factor is squared. This is because area is a two-dimensional measurement.
๐ Historical Context
The concept of scale factors has been used for centuries in mapmaking, architecture, and art. Ancient civilizations understood the importance of maintaining proportions when scaling up or down objects. For example, the Egyptians used grids to scale up drawings for their monumental constructions.
๐ Key Principles
- ๐ Definition of Scale Factor: The ratio by which a figure is enlarged or reduced. If the scale factor is greater than 1, the figure is enlarged. If it's less than 1, the figure is reduced.
- ๐ Area and Scale Factor: If a figure is scaled by a factor of $k$, its area is scaled by a factor of $k^2$.
- โ Formula: If the original area is $A$, and the scale factor is $k$, the new area $A'$ is given by $A' = k^2 \times A$.
โ Solved Problems
Example 1: Scaling a Square
A square has sides of length 5 cm. If we scale the square by a factor of 3, what is the area of the new square?
- ๐ข Original Area: $A = 5 \text{ cm} \times 5 \text{ cm} = 25 \text{ cm}^2$
- ๐ Scale Factor: $k = 3$
- โจ New Area: $A' = k^2 \times A = 3^2 \times 25 \text{ cm}^2 = 9 \times 25 \text{ cm}^2 = 225 \text{ cm}^2$
Example 2: Scaling a Rectangle
A rectangle has a length of 8 cm and a width of 4 cm. If we scale the rectangle by a factor of 0.5, what is the area of the new rectangle?
- ๐ Original Area: $A = 8 \text{ cm} \times 4 \text{ cm} = 32 \text{ cm}^2$
- ๐ Scale Factor: $k = 0.5$
- ๐ก New Area: $A' = k^2 \times A = (0.5)^2 \times 32 \text{ cm}^2 = 0.25 \times 32 \text{ cm}^2 = 8 \text{ cm}^2$
Example 3: Scaling a Circle
A circle has a radius of 2 cm. If we scale the circle by a factor of 2, what is the area of the new circle?
- ๐งฎ Original Area: $A = \pi r^2 = \pi (2 \text{ cm})^2 = 4\pi \text{ cm}^2$
- ๐ Scale Factor: $k = 2$
- โจ New Area: $A' = k^2 \times A = 2^2 \times 4\pi \text{ cm}^2 = 4 \times 4\pi \text{ cm}^2 = 16\pi \text{ cm}^2$
Example 4: Scaling a Triangle
A triangle has a base of 6 cm and a height of 4 cm. If we scale the triangle by a factor of 1.5, what is the area of the new triangle?
- ๐ Original Area: $A = \frac{1}{2} \times 6 \text{ cm} \times 4 \text{ cm} = 12 \text{ cm}^2$
- ๐ Scale Factor: $k = 1.5$
- ๐ก New Area: $A' = k^2 \times A = (1.5)^2 \times 12 \text{ cm}^2 = 2.25 \times 12 \text{ cm}^2 = 27 \text{ cm}^2$
Example 5: Reverse Scaling
A square has an area of 100 cmยฒ. It was created by scaling an original square by a factor of 2. What was the area of the original square?
- ๐ New Area: $A' = 100 \text{ cm}^2$
- ๐ Scale Factor: $k = 2$
- โ Original Area: $A = \frac{A'}{k^2} = \frac{100 \text{ cm}^2}{2^2} = \frac{100 \text{ cm}^2}{4} = 25 \text{ cm}^2$
Example 6: Complex Shapes
A shape composed of two rectangles has a total area of 50 cmยฒ. If the shape is scaled by a factor of 0.8, what is the new total area?
- โ Original Area: $A = 50 \text{ cm}^2$
- ๐ Scale Factor: $k = 0.8$
- โจ New Area: $A' = k^2 \times A = (0.8)^2 \times 50 \text{ cm}^2 = 0.64 \times 50 \text{ cm}^2 = 32 \text{ cm}^2$
Example 7: Real-World Application
An architect is designing a park. The park's area on the blueprint (scale factor 1/1000) is 200 cmยฒ. What will be the actual area of the park in square meters?
- ๐ Blueprint Area: $A = 200 \text{ cm}^2$
- ๐ Scale Factor: $k = 1000$ (since blueprint is scaled down)
- ๐ Actual Area in cmยฒ: $A' = k^2 \times A = (1000)^2 \times 200 \text{ cm}^2 = 1000000 \times 200 \text{ cm}^2 = 200000000 \text{ cm}^2$
- โ Actual Area in mยฒ: Since 1 mยฒ = 10000 cmยฒ, $A' = \frac{200000000 \text{ cm}^2}{10000 \text{ cm}^2/\text{m}^2} = 20000 \text{ m}^2$
๐ก Conclusion
Understanding how scale factors affect area is crucial in various fields, from geometry to real-world applications like architecture and design. Remember, when you scale a figure by a factor of $k$, its area changes by a factor of $k^2$.