Lesson 13
Decomposing Bases for Area
Let’s look at how some people use volume.
13.1: Are These Prisms?
-
Which of these solids are prisms? Explain how you know.
-
For each of the prisms, what does the base look like?
- Shade one base in the picture.
-
Draw a cross section of the prism parallel to the base.
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. Here are the measurements of the base.
![An irregular polygon. Please ask for additional assisstance.](https://staging-cms-im.s3.amazonaws.com/VeFoaUdJbDXYyFAbsNvB64CA?response-content-disposition=inline%3B%20filename%3D%227-7.6.C3.Image.05.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.C3.Image.05.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T055414Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=548aac8b43dddaa04d2313ea1b040665ca6f30060d1d65cb95285a548a308469)
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.](https://staging-cms-im.s3.amazonaws.com/zhUkLWV4zAe15SAQYmcDE2Vy?response-content-disposition=inline%3B%20filename%3D%227-7.6.C3.Image.11v2.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.C3.Image.11v2.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T055414Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ef67f0ac7071297c04360eeb1bd04d9f673e92230910f17e1642fec348ee8ff8)
- 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?
- Between you and your partner, decide which of you will use which of the remaining two methods.
- Using the quadrilaterals and triangles drawn in your selected plan, find the area of the base.
- Trade with a partner and check each other’s work. If you disagree, work to reach an agreement.
-
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?
13.3: 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.](https://staging-cms-im.s3.amazonaws.com/dEPssXJGd1wBMzQR97ESAo3W?response-content-disposition=inline%3B%20filename%3D%227-7.6.C3.Image.14.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.C3.Image.14.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T055414Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c075c171fbdc16b353501e72033f51558a7c181849f4f304f714e0dc51e7e7de)
-
Draw the base of this prism and label its dimensions.
-
What is the area of the base? Explain or show your reasoning.
- What is the volume of the prism?
Summary
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.](https://staging-cms-im.s3.amazonaws.com/bJhPjt5yAhH9QZ4ciBJvhRiv?response-content-disposition=inline%3B%20filename%3D%227-7.6.C3.Image.09.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.C3.Image.09.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T055414Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=8f6d23ed292d2c92fd44d89ff809adb9a68de9eb9654dfbba2b5a0e8b6191105)
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.](https://staging-cms-im.s3.amazonaws.com/WoVQ1kF7zD1q48aUVSYo9L5s?response-content-disposition=inline%3B%20filename%3D%227-7.6.C3.Image.10.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.C3.Image.10.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T055414Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a66d98855a2df1823e375f9a3e1f5b12f064e63c8db511c097e7dfed0f1da58c)
\(\displaystyle V = Bh\)
Video Summary
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.
- 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.
- 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.
- 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.
- 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.