Does Bigger Perimeter Mean Bigger Area?
Both diagrams represent models for a backyard patio. Both diagrams use the same number of identical concrete pieces (not drawn to scale). The area of each patio is the same, 180 square meters. What are the dimensions of a single rectangle piece of concrete?
Hints:
- Notice how the pieces fit together in the second model. What is different about the way they fit together in the first model?
- The formula to find the area of a rectangle is A=l*w
- Rectangles with different perimeters can have the same area; don’t let the pictures fool you!
- Try setting up a ratio and then do a substitution.
Solution:
Complete Solution:
Each tile is 4 meters by 5 meters.
The area of each patio is 180 square meters (m2), and each is made from nine identical rectangular concrete pieces. This means that the area of one piece is 180 ÷ 9, or 20 m2.
Since the area of the rectangle equals length (L) times width (W), you know that L × W = 20. From the way in which the pieces are arranged in the second patio, you can see that four lengths is the same as five widths. In other words, the ratio of length to width is 5 to 4.
As it happens, 5 × 4 = 20. This means that the length is 5m, while the width is 4m.
Alternate Solution:
Another way to look at this is as follows. You know from the second patio that four lengths equals five widths. Since 4L = 5W, then:
L = 5/4 × W
You also know the area of the piece:
L × W = 20. So, substituting for L,
(5/4 × W) × W = 20
5/4 W2 = 20
W2 = 4/5 × 20
W2 = 16
W = 4 (Widths cannot be negative.)
Substituting 4 for W, you can then determine that L = 5. So a single concrete piece has dimensions 4m by 5m.
Solution:
Each tile is 4 meters by 5 meters.
Each tile is 4 meters by 5 meters.
The area of each patio is 180 square meters (m2), and each is made from nine identical rectangular concrete pieces. This means that the area of one piece is 180 ÷ 9, or 20 m2.
Since the area of the rectangle equals length (L) times width (W), you know that L × W = 20. From the way in which the pieces are arranged in the second patio, you can see that four lengths is the same as five widths. In other words, the ratio of length to width is 5 to 4.
As it happens, 5 × 4 = 20. This means that the length is 5m, while the width is 4m.
Alternate Solution:
Another way to look at this is as follows. You know from the second patio that four lengths equals five widths. Since 4L = 5W, then:
L = 5/4 × W
You also know the area of the piece:
L × W = 20. So, substituting for L,
(5/4 × W) × W = 20
5/4 W2 = 20
W2 = 4/5 × 20
W2 = 16
W = 4 (Widths cannot be negative.)
Substituting 4 for W, you can then determine that L = 5. So a single concrete piece has dimensions 4m by 5m.
Solution:
Each tile is 4 meters by 5 meters.