Cross-Sections and Scale Factors

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• Three dimensional (3D): having length, width, and height, taking up space in all directions

• Cross section: a flat shape created by slicing a 3D object

• Scale factor: a number that represents how many times bigger or smaller a shape or object is than its original size

Add these to your Definitions section

Add these to your Properties section

Cross-Sections

Different solids can be cut at different angles to create different cross sections. Remember that a cross section is a flat shape, not a solid object or a line. The image shows some possible cross-sections of a cylinder.

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Draw or describe three possible cross sections of this pyramid.

Open Ended

List three solids that have circular cross sections.

Exploring Scale Factors

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Predict: how many 1-inch squares would fit inside a 2-inch square?

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Predict: how many 1-inch squares would fit inside a 2-inch square?

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Predict: how many 1-inch squares would fit inside a 3-inch square?

For each square, the side lengths are 2, 3 and 4 times bigger than the original 1-inch square

• The 2-inch square fits 4 squares

• The 3-inch square fits 9 squares

• The 4-inch square fits 16 squares

Instead of being 2, 3, or 4 times bigger than the original area, the areas are 4, 9, and 16 times bigger.

The area scaled up by the original scale factor squared.

Exploring Scale Factors

Open Ended

Predict: how many 1-inch cubes will it take to make a 2-inch cube?

Open Ended

Predict: how many 1-inch cubes will it take to make a 3-inch cube?

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Predict: how many 1-inch cubes will it take to make a 4-inch cube?

For each cube, the side lengths are 2, 3 and 4 times bigger than the original 1-inch cube

• The 2-inch cube needs 8 cubes

• The 3-inch square needs 27 cubes

• The 4-inch square needs 64 cubes

Instead of being 2, 3, or 4 times bigger than the original area, the volumes are 8, 27, and 64 times bigger.

The volume scaled up by the original scale factor cubed.

Add these properties to your notebook

Example: Scaling Area and Volume

Example: Scaling Area and Volume

Multiple Choice

A pyramid with volume $$27\ in^3$$ and surface area $$63\ in^2$$ is dilated by a scale factor of 0.5. Will the new area and volume be bigger, smaller, or the same as the original?

• 1. Bigger, because the scale factor is bigger than 1

• 1. Smaller, because the scale factor is smaller than 1

• 1. The same, because the scale factor is 1

Open Ended

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