Lesson 11
Polyhedra and Nets
Let's use nets to find the surface area of polyhedra.
Problem 1
Here are two three-dimensional figures.
Tell whether each of the following statements describes Figure A, Figure B, both, or neither.
![Figure A triangular prism. Figure B triangular pyramid.](https://cms-im.s3.amazonaws.com/GMeWfd8yn4mH4Hm3EvM9TK2Y?response-content-disposition=inline%3B%20filename%3D%226-6.1.E.PP_Image_4.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.E.PP_Image_4.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T010006Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c295a25c426f81e326b42d694038f25e9c66eb63c3a966546a71c11c0ef77244)
- This figure is a polyhedron.
- This figure has triangular faces.
- There are more vertices than edges in this figure.
- This figure has rectangular faces.
- This figure is a pyramid.
- There is exactly one face that can be the base for this figure.
- The base of this figure is a triangle.
- This figure has two identical and parallel faces that can be the base.
Problem 2
-
Is this polyhedron a prism, a pyramid, or neither? Explain how you know.
- How many faces, edges, and vertices does it have?
Problem 3
-
What polyhedron can be assembled from this net? Explain how you know.
- Find the surface area of this polyhedron. Show your reasoning.
Problem 4
-
A parallelogram has a base of 12 meters and a height of 1.5 meters. What is its area?
- A triangle has a base of 16 inches and a height of \(\frac18\) inches. What is its area?
- A parallelogram has an area of 28 square feet and a height of 4 feet. What is its base?
- A triangle has an area of 32 square millimeters and a base of 8 millimeters. What is its height?
Problem 5
Find the area of the shaded region. Show or explain your reasoning.
![Triangle shaded with a unshaded square located in the middle on the bottom.](https://cms-im.s3.amazonaws.com/J1qEMcijroqnTp77iHX5Z1ZH?response-content-disposition=inline%3B%20filename%3D%226-6.1.A.PP_Image_17.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A.PP_Image_17.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T010006Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=544954cc5bec6e2c652da90439ce5c595b9465171dc3374a18dc15fedd51e0a8)