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How do you relate scale drawings to area?

Published in Scale Drawings 2 mins read

Scale drawings are related to area because when you scale a drawing, the area changes by the square of the scale factor.

Here's a breakdown of how scale drawings relate to area:

  • Scale Factor: A scale drawing represents an object proportionally, either larger or smaller than its original size. The scale factor is the ratio that defines this proportion. For example, a scale factor of 2 means the drawing is twice the size of the original, and a scale factor of 0.5 (or 1/2) means the drawing is half the size of the original.

  • Area and Scaling: When you scale a two-dimensional figure, you're essentially multiplying both its length and width by the scale factor. This means that the area, which is length times width, gets multiplied by the scale factor twice.

  • The Formula: The relationship between the area of a scale drawing (As) and the actual area (A) can be expressed as:

    *A = s2 As**

    Where:

    • A = Actual Area
    • s = Scale Factor
    • As = Area of the Scale Drawing

Example:

Let's say you have a rectangular garden that's represented by a scale drawing with a length of 5 inches and a width of 3 inches. The scale of the drawing is 1 inch = 2 feet.

  1. Area of the Scale Drawing (As):

    As = length width = 5 inches 3 inches = 15 square inches

  2. Scale Factor (s):

    Since 1 inch on the drawing represents 2 feet in reality, the scale factor is 2.

  3. Actual Area (A):

    A = s2 As = 22 15 square inches = 4 15 square feet = 60 square feet. Notice that the actual area uses feet as its area unit. Because each dimension of the scale drawing represents feet.
    So, 5 inches represents 10 feet, and 3 inches represents 6 feet. The actual area of the garden would be 10 feet     6 feet = 60 square feet.

In summary, when working with scale drawings and area, remember that the area changes by the square of the scale factor. If you double the dimensions, you quadruple the area. If you triple the dimensions, you multiply the area by nine, and so on. This understanding is crucial for accurately interpreting scaled representations in fields like architecture, engineering, and cartography.

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