Question
Find the perimeter and the area of the polygon shape shown. All measurements are in inches.
An image shows a plus-shaped polygon with a square connected at the center of the top length of a rectangle and a square connected at the center of the bottom length of the rectangle. The squares have sides of 2 inches and the rectangle is 2 inches wide and 6 inches long.
An image shows a plus-shaped polygon with a square connected at the center of the top length of a rectangle and a square connected at the center of the bottom length of the rectangle. The squares have sides of 2 inches and the rectangle is 2 inches wide and 6 inches long.
Answers
GPT-4o mini
To find the perimeter and area of the plus-shaped polygon, we need to analyze the given information.
### Dimensions:
- The squares have sides of **2 inches**.
- The rectangle has a width of **2 inches** and a length of **6 inches**.
### Shape Description:
The plus-shaped polygon consists of:
1. A rectangle that is **2 inches wide and 6 inches long**.
2. A square on the top, centered at the top side of the rectangle.
3. A square on the bottom, centered at the bottom side of the rectangle.
### 1. **Calculating the Perimeter:**
To find the perimeter, we need to add up all the outer sides of the plus-shaped figure.
- The sides of the rectangle can be categorized as follows:
- The left and right sides of the rectangle: Since the rectangle is 2 inches wide, both sides contribute **2 inches** each. Thus, the contribution from the two sides is \(2 + 2 = 4\) inches.
- The top and bottom sides of the rectangle are **6 inches** each, but since they are shared with the squares on top and bottom, we need to count only the segments that are outside the squares.
The squares have sides of 2 inches, so the contribution from the top and bottom sides is:
- For the top side: we take the outer segments of the rectangle (left portion of rectangle + width of square on the right):
- Left horizontal segment of rectangle: **2 inches**
- Width of square: **2 inches**
- Right horizontal segment of rectangle (it is flushed with the square): **2 inches**
Total from the top side = \(2 + 2 = 4\) inches (both parts are contributing as squares overlap with the rectangle)
The same calculation applies to the bottom side which contributes another 4 inches.
So the total perimeter can be calculated as:
\[
P = (2 + 2) + (2 + 2 + 2 + 2) + (2 + 2) = 4 + 4 + 4 = 12 \text{ inches}
\]
### 2. **Calculating the Area:**
To find the area, we can separately calculate the area of the rectangle and the area of the two squares, then sum them up.
- **Area of the Rectangle:**
\[
\text{Area of rectangle} = \text{width} \times \text{length} = 2 \times 6 = 12 \text{ square inches}
\]
- **Area of Each Square:**
\[
\text{Area of each square} = \text{side} \times \text{side} = 2 \times 2 = 4 \text{ square inches}
\]
Since there are **two squares**, we get:
\[
\text{Total area from squares} = 2 \times 4 = 8 \text{ square inches}
\]
### Total Area Calculation:
\[
\text{Total Area} = \text{Area of rectangle} + \text{Total area from squares} = 12 + 8 = 20 \text{ square inches}
\]
### Conclusion:
- **Perimeter**: \(12\) inches
- **Area**: \(20\) square inches
### Dimensions:
- The squares have sides of **2 inches**.
- The rectangle has a width of **2 inches** and a length of **6 inches**.
### Shape Description:
The plus-shaped polygon consists of:
1. A rectangle that is **2 inches wide and 6 inches long**.
2. A square on the top, centered at the top side of the rectangle.
3. A square on the bottom, centered at the bottom side of the rectangle.
### 1. **Calculating the Perimeter:**
To find the perimeter, we need to add up all the outer sides of the plus-shaped figure.
- The sides of the rectangle can be categorized as follows:
- The left and right sides of the rectangle: Since the rectangle is 2 inches wide, both sides contribute **2 inches** each. Thus, the contribution from the two sides is \(2 + 2 = 4\) inches.
- The top and bottom sides of the rectangle are **6 inches** each, but since they are shared with the squares on top and bottom, we need to count only the segments that are outside the squares.
The squares have sides of 2 inches, so the contribution from the top and bottom sides is:
- For the top side: we take the outer segments of the rectangle (left portion of rectangle + width of square on the right):
- Left horizontal segment of rectangle: **2 inches**
- Width of square: **2 inches**
- Right horizontal segment of rectangle (it is flushed with the square): **2 inches**
Total from the top side = \(2 + 2 = 4\) inches (both parts are contributing as squares overlap with the rectangle)
The same calculation applies to the bottom side which contributes another 4 inches.
So the total perimeter can be calculated as:
\[
P = (2 + 2) + (2 + 2 + 2 + 2) + (2 + 2) = 4 + 4 + 4 = 12 \text{ inches}
\]
### 2. **Calculating the Area:**
To find the area, we can separately calculate the area of the rectangle and the area of the two squares, then sum them up.
- **Area of the Rectangle:**
\[
\text{Area of rectangle} = \text{width} \times \text{length} = 2 \times 6 = 12 \text{ square inches}
\]
- **Area of Each Square:**
\[
\text{Area of each square} = \text{side} \times \text{side} = 2 \times 2 = 4 \text{ square inches}
\]
Since there are **two squares**, we get:
\[
\text{Total area from squares} = 2 \times 4 = 8 \text{ square inches}
\]
### Total Area Calculation:
\[
\text{Total Area} = \text{Area of rectangle} + \text{Total area from squares} = 12 + 8 = 20 \text{ square inches}
\]
### Conclusion:
- **Perimeter**: \(12\) inches
- **Area**: \(20\) square inches