Lesson 3
Reasoning to Find Area
Let’s decompose and rearrange shapes to find their areas.
3.1: Comparing Regions
Is the area of Figure A greater than, less than, or equal to the area of the shaded region in Figure B? Be prepared to explain your reasoning.
![Square A, shaded. Square B identical to A, with a small shaded square removed in the middle and a small shaded square appended to its side.](https://cms-im.s3.amazonaws.com/etMDgYdD51AJJ2HCr6eDW6jC?response-content-disposition=inline%3B%20filename%3D%226-6.1.A3_Image_1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A3_Image_1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T235304Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=72f5a41d42e39d055822dc74f25f2af1cdf00367f3f7a62f812d8376884e2387)
3.2: On the Grid
Each grid square is 1 square unit. Find the area, in square units, of each shaded region without counting every square. Be prepared to explain your reasoning.
![Four figures, each on a white square grid.](https://cms-im.s3.amazonaws.com/qzFvtghh4zzdnxghxeLd6aQA?response-content-disposition=inline%3B%20filename%3D%226-6.1.A3_Image_2.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A3_Image_2.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T235304Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e92668eeff625587c99ef0911138b4de466a0cf3398bac58235542bea92e4340)
Rearrange the triangles from Figure C so they fit inside Figure D. Draw and color a diagram of your work.
3.3: Off the Grid
Find the area of the shaded region(s) of each figure. Explain or show your reasoning.
![3 figures labeled A, B, C.](https://cms-im.s3.amazonaws.com/Barv3HtKUXJYULZi85GZywoz?response-content-disposition=inline%3B%20filename%3D%226-6.1.A3_Image_7.1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A3_Image_7.1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T235304Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=45277549c3175c3c20172a0ebb2ef900e8fbe12e81af8ab6674e729607ae051e)
Summary
There are different strategies we can use to find the area of a region. We can:
- Decompose it into shapes whose areas you know how to calculate; find the area of each of those shapes, and then add the areas.
![Two images of t-shaped objects on a grids.](https://cms-im.s3.amazonaws.com/coATzR5ewE6kmsmm3uESTnB1?response-content-disposition=inline%3B%20filename%3D%226-6.1.A3.Image.09a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A3.Image.09a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T235304Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c5df1271088719239ccea57138559f3908f41a86b6844be715d7e4bdaa2239ea)
- Decompose it and rearrange the pieces into shapes whose areas you know how to calculate; find the area of each of those shapes, and then add the areas.
![3 figures on grids with arrows pointing to the right between figures 1 and 2 and figures 2 and 3.](https://cms-im.s3.amazonaws.com/y77zc1jAWayjr6FRZgTSuM4B?response-content-disposition=inline%3B%20filename%3D%226-6.1.A3.Image.10a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A3.Image.10a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T235304Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=3b219bfe6d7e81c0f3238f8d0aa21ab38664a2fce3c721aa2c3d7eb4d8e37994)
- Consider it as a shape with a missing piece; calculate the area of the shape and the missing piece, and then subtract the area of the piece from the area of the shape.
![Two shaded squares in a grid. Each are 6 units square and each as a 1 unit by two unit portion that is unshaded.](https://cms-im.s3.amazonaws.com/wtKDoWUzEBFQU35VKLRrPGf8?response-content-disposition=inline%3B%20filename%3D%226-6.1.A3.Image.11a.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A3.Image.11a.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T235304Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=27ee4c923c7187379877613a4c93d16bd50c2b5b9b1377d0055f749a9cdcf010)
The area of a figure is always measured in square units. When both side lengths of a rectangle are given in centimeters, then the area is given in square centimeters. For example, the area of this rectangle is 32 square centimeters.
![rectangle, base = 8 centimeters, height = 4 centimeters.](https://cms-im.s3.amazonaws.com/h8hN4SGpgFvHar497szZP83r?response-content-disposition=inline%3B%20filename%3D%226-6.1.A3_Image_12.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.A3_Image_12.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T235304Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9320e28a08aee09908a2384ad0b4654da423198be562417fbe4da16a59f93262)
Video Summary
Glossary Entries
- area
Area is the number of square units that cover a two-dimensional region, without any gaps or overlaps.
For example, the area of region A is 8 square units. The area of the shaded region of B is \(\frac12\) square unit.
- compose
Compose means “put together.” We use the word compose to describe putting more than one figure together to make a new shape.
- decompose
Decompose means “take apart.” We use the word decompose to describe taking a figure apart to make more than one new shape.
- region
A region is the space inside of a shape. Some examples of two-dimensional regions are inside a circle or inside a polygon. Some examples of three-dimensional regions are the inside of a cube or the inside of a sphere.