Lesson 15

Decomposing Bases for Area

Let’s look at how some people use volume.

15.1: Are These Prisms?

  1. Which of these solids are prisms? Explain how you know.

    A collection of 6 3-dimensional shapes.  Please ask for additional assisstance.
  2. For each of the prisms, what does the base look like?

    1. Shade one base in the picture.
    2. Draw a cross section of the prism parallel to the base.

15.2: 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. Here are the measurements of the base.

An irregular polygon.  Please ask for additional assisstance.

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.

Three images copies of the previous irregular polygon. Lin's, decomposed into triangles and trapezoids, Jada's decomposed into triangles and rectangles, Diego's supplemented to form a rectangle.
  1. 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. 
  2. Although all three methods could work, one of them requires measurements that are not provided. Which one is it?
  3. Between you and your partner, decide which of you will use which of the remaining two methods. 
  4. Using the quadrilaterals and triangles drawn in your selected plan, find the area of the base. 
  5. Trade with a partner and check each other’s work. If you disagree, work to reach an agreement. 
  6. Return their work. Calculate the volume of the box of chocolates.

The box has 30 pieces of chocolate in it, each with a volume of 1 in3. If all the chocolates melt into a solid layer across the bottom of the box, what will be the height of the layer?

15.3: Another Prism

A house-shaped prism is created by attaching a triangular prism on top of a rectangular prism.

A prism. Base, pentagon. The pentagon is a 7 by 6 rectangle with a triangle on top that has sides 6, 5, 5. The total height of the pentagon is 11. The prism has height 8.
  1. Draw the base of this prism and label its dimensions.

  2. What is the area of the base? Explain or show your reasoning.

  3. What is the volume of the prism?


To find the area of any polygon, you can decompose it into rectangles and triangles. There are always many ways to decompose a polygon.

Four images of the same irregular polygon.  In two images, the polygon is cut into different triangles and rectangles.  In the fourth image, a triangle is added to make the polygon a rectangle.

Sometimes it is easier to enclose a polygon in a rectangle and subtract the area of the extra pieces.

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, \(h\).

 A prism.  The base of the prism is the irregular polygon from the previous images, area B, and the prism has height h.

\(\displaystyle V = Bh\)

Glossary Entries

  • base (of a prism or pyramid)

    The word base can also refer to a face of a polyhedron.

    A prism has two identical bases that are parallel. A pyramid has one base.

    A prism or pyramid is named for the shape of its base.

    Two figures, a pentagonal prism and a hexagonal pyramid.
  • cone

    A cone is a three-dimensional figure like a pyramid, but the base is a circle.

  • cross section

    A cross section is the new face you see when you slice through a three-dimensional figure.

    For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.

  • cylinder

    A cylinder is a three-dimensional figure like a prism, but with bases that are circles.

  • prism

    A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.

    Here are some drawings of prisms.

    A triangular prism, a pentagonal prism, and a rectangular prism.
  • pyramid

    A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.

    Here are some drawings of pyramids.

    a rectangular pyramid, a hexagonal pyramid, a heptagonal pyramid
  • sphere

    A sphere is a three-dimensional figure in which all cross-sections in every direction are circles.

  • volume

    Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.

    For example, the volume of this rectangular prism is 60 units3, because it is composed of 3 layers that are each 20 units3.

    Two images. First, a prism made of cubes stacked 5 wide, 4 deep, 3 tall. Second, each of the layers of the prism is separated to show 3 prisms 5 wide, 4 deep, 1 tall.