Effects of Changing Dimensions on Perimeter, Area & Volume


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When you change the dimensions of a geometric figure, its area and volume are affected in a specific, predictable way, provided all linear dimensions are changed proportionally (meaning the new figure is similar to the original). This proportional change is described by a linear scale factor.

The following diagram gives the effects of changing dimensions proportionally on the area, perimeter, and volume of figures. Scroll down the page for more examples and solutions.
Effects of Changing Dimensions on Perimeter, Area, and Volume
 

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The Linear Scale Factor (k)
Let k be the factor by which all linear dimensions of a figure are scaled.
If k>1, the figure is enlarged.
If 0<k<1, the figure is reduced.
If k=1, the figure remains the same size.

Effect on Area (2D figures and Surface Area of 3D figures)
Area is a two-dimensional measurement, expressed in square units. When all linear dimensions of a figure are scaled by a factor of k, its area changes by a factor of k2.
New Area = Original Area × k2.

Effect on Volume (3D figures)
Volume is a three-dimensional measurement, expressed in cubic units. When all linear dimensions of a figure are scaled by a factor of k, its volume changes by a factor of k3.

Summary of Scaling Relationships
When all linear dimensions of a similar figure are scaled by a factor of k:
Linear dimensions (e.g., perimeter, circumference, height, radius, side length): change by a factor of k
Area (e.g., 2D area, surface area): change by a factor of k2.
Volume (e.g., 3D volume): change by a factor of k3.

Effects of Changing Dimensions Proportionally
Effects of Changing One Dimension
Examples:

  1. Describe the effect of the change on the area when the height of a parallelogram is doubled.
  2. Describe the effect of the change on the area when the base length of a triangle is halved.

Effects of Changing Dimensions Proportionally
Examples:

  1. Describe the effect of the change on the perimeter and area when the base and height of a rectangle with base 8 m and height 3 m are both multiplied by 5.
  2. Describe the effect of the change on the circumference and area of a circle when the radius is multiplied by 1/3.

Effects of Changing Area
Examples:

  1. A square has side length 5 cm. If the area is tripled, what happens to the side length?
  2. A circle has a radius of 6 in. If the area is doubled, what happens to the circumference?



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