Effects of Changing Dimensions on Perimeter, Area & Volume

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.

 

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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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