Finding the Area of an Original Geometric Design

Question

For this activity you will need an 8 1/2x11 inch sheet of paper, a pencil, and a ruler.

1. Create an original design using only the figures given. The side lengths of all of the shapes are the same so that you can line up their edges to create a tiling. Your design can use any combination of shapes, but you must use at least one of each shape in your design. The design you create can be as simple or as complex as you want. It can have a repeating pattern or it can be random, but it must fit on a single sheet of paper. You can print and us the shapes provided in the image below or measure and draw your own shapes. Your design can make a recognizable shape or it can be abstract. Use your imagination!

2. Think about the shapes you used in your design and how you will calculate the area of each. Are these any ways you can make it easier to find the area?

3. Use a ruler to make any measurements you need to find the area of each shape. Refer to your formulas to find out which measurements you will need. Record your measurements in a table.

4. Calculate the area of each shape you used. Find the area of at least one shape by composing a rectangle. Find the area of at least one shape by decomposing it into simpler shapes. Record your calculations in a table.

5. Calculate the total area of the figure you designed. Show your work. Include the correct units for area in your answer.

6. Describe in words how you calculated the area of each shape. Be sure to include which formulas you used. Did you compose a rectangle to find the area or did you decompose the shape into simpler polygons? Write it as a 6th grader

Answer 1

Step 1: Creating My Design

I took an 8 1/2 x 11 inch sheet of paper and, using my pencil and ruler, I created an original geometric design. For my design, I used the following shapes:

• Squares
• Rectangles
• Triangles
• Circles

I made a pattern where I combined these shapes in interesting ways, making sure they aligned perfectly to create a nice tiling effect.

Step 2: Planning to Calculate Areas

To find the area of each shape, I thought about how I could easily calculate them. Here are some tips:

• For squares and rectangles, I can use the formula: Area = length × width.
• For triangles, the formula is: Area = 1/2 × base × height.
• For circles, the area can be found using: Area = π × radius².

I decided to use compositions (combining shapes) and decompositions (breaking shapes into simpler ones) to make my calculations easier.

Step 3: Measuring the Shapes

I measured my shapes with a ruler. Here’s a table of my measurements:

| Shape | Measurement 1 | Measurement 2 | Shape Type | |-------------|---------------|---------------|--------------------| | Square | 2 inches | 2 inches | Square | | Rectangle | 4 inches | 2 inches | Rectangle | | Triangle | Base: 4 inches | Height: 2 inches | Triangle | | Circle | Radius: 1 inch | | Circle |

Step 4: Calculating Areas of Each Shape

Now, I will calculate the area using the formulas I noted.

1. Square:

   • Area = length × width = 2 inches × 2 inches = 4 square inches.

2. Rectangle:

   • Area = length × width = 4 inches × 2 inches = 8 square inches.

3. Triangle:

   • Area = 1/2 × base × height = 1/2 × 4 inches × 2 inches = 4 square inches.

4. Circle:

   • Area = π × radius² = π × (1 inch)² ≈ 3.14 × 1 = 3.14 square inches.

Step 5: Total Area of the Design

Now, I will calculate the total area of my design by adding up the areas of all the shapes I used:

• Square: 4 square inches
• Rectangle: 8 square inches
• Triangle: 4 square inches
• Circle: 3.14 square inches

Total Area = 4 + 8 + 4 + 3.14 = 19.14 square inches.

So, the total area of my geometric design is 19.14 square inches.

Step 6: Describing How I Calculated Area

To calculate the area of each shape, I used different formulas based on the shape:

• For the square, I multiplied the length (2 inches) by the width (2 inches) to get an area of 4 square inches.
• For the rectangle, I used its length (4 inches) and width (2 inches) and found the area to be 8 square inches.
• The triangle was calculated by taking half of the base (4 inches) multiplied by the height (2 inches), resulting in 4 square inches.
• Lastly, for the circle, I used the radius (1 inch) in the formula π × radius², which gave me approximately 3.14 square inches.

I found it helpful to use both composition and decomposition. The square and rectangle were straightforward, while I clearly broke down the triangle and used a specific formula for the circle. Overall, calculating the areas was fun and made me appreciate how shapes can fit together!