suppose one had 100 yards of fencing to enclose a garden. What shapes could be enclosed? What are the dimensions of each and what is the area? make a chart.
Wow
rectangles?
squares?
triangles?
hexagons?
....
star-shaped things
convex septagons ?
...
To explore the possible shapes that can be enclosed with 100 yards of fencing, we need to consider various geometric shapes and calculate their respective dimensions and areas. Here's a chart that includes some common shapes:
| Shape | Dimensions | Formula for Perimeter | Perimeter | Formula for Area | Area |
|----------|------------------------|-----------------------|-----------|-----------------|-----------------|
| Square | Length = Width | P = 4s | ? | A = s^2 | ? |
| Rectangle | Length > Width | P = 2L + 2W | ? | A = L * W | ? |
| Circle | Radius = Diameter/2 | P = 2πr | ? | A = πr^2 | ? |
| Triangle | a, b, c represent sides | P = a + b + c | ? | A = ½ * b * h | ? |
Let's calculate the missing values in the table:
For a square: Each side has a length of (100 / 4) = 25 yards. So, the perimeter is 4 * 25 = 100 yards, and the area is 25 * 25 = 625 square yards.
For a rectangle: The possibilities for length and width can be many, so we need to find combinations where the perimeter is equal to 100 yards. For example, with a length of 10 yards, the width would be (100 - 2 * 10) / 2 = 40 yards. This yields a perimeter of (2 * 10) + (2 * 40) = 100 yards and an area of 10 * 40 = 400 square yards. There could be other valid rectangles with different dimensions but still 100 yards of fencing.
For a circle: The circumference of a circle is given by 2πr, which should equal 100 yards. Solving for the radius 'r', we have r = (100 / (2π)) ≈ 15.92 yards. Therefore, the perimeter is (2 * π * 15.92) ≈ 100 yards, and the area is (π * (15.92^2)) ≈ 797.78 square yards.
For a triangle: We need to consider combinations of side lengths that form a perimeter of 100 yards. For example, a triangle with sides measuring 30, 35, and 35 yards (isosceles triangle) would have a perimeter of (30 + 35 + 35) = 100 yards. To calculate the area, we need the height of the triangle. If we consider the base of 30 yards, we can drop a perpendicular line from the opposite angle to the base, forming two equal right-angled triangles. Using the Pythagorean theorem, we find the height is √(35^2 - (30/2)^2) ≈ 23.43 yards. Hence, the area is (½ * 30 * 23.43) ≈ 351.45 square yards.
Please note that there may be other valid shapes and dimensions that satisfy the given conditions.