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=20240726T235647Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ba218aa1e65286d77677c2901a8b4851e0126fb418666b176c08902844d0eba6)
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=20240726T235647Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=526d86ecb64975908dde80bf40b191fa6474507b0123bc976579e3ba527271ee)
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=20240726T235648Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=c4b956606d6ad9b8e00e1649af71a241c76af37bf61ce3e6d45388e23c7ec9e6)
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=20240726T235648Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=fafb78855ad9a0d27451d90ef38a2eb7566ba2bcb0ebf3f7cce336c2f8d00b9e)
- 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=20240726T235648Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=954a790b2e9219350225907b0ffb5c07db47fd0265b88ee896dfc28417bb4b6f)
- 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=20240726T235648Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f7e560a5ccd569e58609d2531153c84c073d55961d836e1c091f755a3d5645a3)
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=20240726T235648Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=fc3d1e7999973bdc52cf6c55f053a1f7d67aad499608e38ffb1e949160ff901f)
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.