Understanding how to calculate areas and volumes in similar shapes is essential for solving geometry problems efficiently. This section focuses on using scale factors to relate areas and volumes between similar shapes.

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Introduction to Scale Factors

Similar shapes maintain the same shape but vary in size. The scale factor indicates the ratio of any two corresponding lengths in these shapes. Key points include:

• Length scale factor: If the scale factor is $k$, lengths in the similar shape are $k$ times those in the original.
• Area scale factor: Equals the square of the length scale factor $(k^2)$.
• Volume scale factor: Equals the cube of the length scale factor $(k^3)$.

Understanding Area Scale Factor

The area scale factor shows how the area of one shape compares to the area of a similar shape.

Example 1: Calculating Area of Similar Shapes

Given two similar rectangles where the second is three times as long as the first. If the first rectangle's area is $15 \text{ cm}^2$, find the area of the second rectangle.

• Given:
  • Length scale factor = 3
  • Area of the first rectangle = $15 \text{ cm}^2$
• Calculate Area Scale Factor: $\text{Area scale factor} = 3^2 = 9$
• Calculate Area of Second Rectangle: $\text{Area of second rectangle} = 15 \times 9 = 135 \text{ cm}^2$

This illustrates that the area of the second rectangle is nine times the area of the first rectangle.

Understanding Volume Scale Factor

The volume scale factor is crucial when comparing the volumes of similar shapes.

Example 2: Calculating Volume of Similar Shapes

Consider two similar spheres, with the second sphere's radius twice the first's. If the first sphere's volume is $36 \text{ cm}^3$, find the volume of the second sphere.

• Given:
  • Length scale factor = 2
  • Volume of the first sphere = $36 \text{ cm}^3$
• Calculate Volume Scale Factor: $\text{Volume scale factor} = 2^3 = 8$
• Calculate Volume of Second Sphere: $\text{Volume of the second sphere} = 36 \times 8 = 288 \text{ cm}^3$

Doubling the radius results in an eightfold increase in volume.

Practical Applications

Applying these principles helps solve real-world problems efficiently.

Example 3: Real-world Application

An architect is creating a scale model of a building with a 1:50 scale factor. If the model's volume is $1.44 \text{ m}^3$, find the volume of the actual building.

• Given:
  • Length scale factor = 50
  • Volume of the model = $1.44 \text{ m}^3$
• Calculate Volume Scale Factor: $\text{Volume scale factor} = 50^3 = 125,000$
• Calculate Volume of the Building: $\text{Volume of the building} = 1.44 \times 125,000 = 180,000 \text{ m}^3$