~ 7.13 ~

Decomposing Bases for Area

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Learning Targets

I can calculate the volume of a prism with a complicated base by decomposing the base into quadrilaterals or triangles.

Notes

To find the area of any polygon, you can decompose it into rectangles and triangles. There are always many ways to decompose a polygon. To decompose a shape means to break it apart into other smaller (and sometimes different) shapes. To compose a figure means to put together shapes to form the new figure. By composing or decomposing, you can more readily take a step-by-step approach to determining formulas for area, perimeter, and volume.

Similarly, sometimes it is easier to enclose a polygon in a rectangle and subtract the area of the decomposed shapes.

Know These Formulas!

To find the volume of a prism with a polygon for a base, you find the area of the base, B, and multiply by the height, h.

Beware, sometimes they flip the prism so that height is laying down instead of going up.

V = B h

Activities

13.2 A Box of Chocolates

A box of chocolates is a prism with a base in the shape of a heart and a height of 2 inches. Below are the measurements of the base.

To calculate the volume of the box, three different students have each drawn line segments showing how they plan on finding the area of the heart-shaped base.

For each student’s plan, describe the shapes the student must find the area of and the operations they must use to calculate the total area.
Although all three methods could work, one of them requires measurements that are not provided. Which one is it?
Using the quadrilaterals and triangles drawn in your selected plan, find the area of the base.
Calculate the volume of the box of chocolates.

We will encounter figures that have non-rectangular bases in future activities or lessons. It will be important to think about different strategies to calculate the area of the base.

13.4 Volume of a Pentagonal Prism

Prism with Pentagonal Base

Two-Dimensional View of the Base

Here is a prism with a pentagonal base. The height is 8 cm. What is the volume of the prism? Show your thinking. Organize it so it can be followed by others.

Add to Your Notes

When the base is not a rectangle or a triangle, what are some methods for finding the area?

Start by calculating the area of the pentagonal base by decomposing that figure into shapes that you know how to find the area of ...rectangles, triangles, trapezoids...

The Pentagon, Arlington County, Virgina

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