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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T002306Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c0e4bcda2fa4fdf57397bd26b515607d35383a90160b848c33354cdee6ac41b9)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T002306Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=81f608c27423c0c534f71158ca152e108bd1efcd12df93c4a9aff89e0c6ae136)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T002306Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b29fccae9391ed2f805fdb5c43c463ca17b9c3d4c5413b0127843fbee534ea88)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T002306Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=0be3337aa6c2bf607c1609480d0c987e4b92dad6d09e2055e176bcc0eb3652fc)
- 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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T002306Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e609132aba58ef67db3b01f13d8f4548d9e8698f1f6ae902dd2b0a88d1cf4888)
- 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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T002306Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c89d38bb0042e23f1ee6321bd1683b479a14545a75c4626b9509d67f71dd7c83)
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%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T002306Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=4407ced114ddc9e0d7a0cce66361621eb389654470a489b86bcab919456b5f97)
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.