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Area & Perimeter of Complex Shapes
Examples, solutions, videos, worksheets, and stories to help Grade 5 students learn the effects of changing dimensions on area and perimeter.
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:
Effects of Changing Dimensions Proportionally
Examples:
Effects of Changing Area
Examples:
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