what is the area of the lagrest circle that can be cut from a rectangular piece of construction paper that is 8 1/2 by 11 inches.
The diameter must be 8.5 inches
A = pi * r^2
A = 3.14 * 4.25^2
A = 3.14 * 18.0625
A = ?
To find the largest circle that can be cut from a rectangular piece of construction paper, we need to determine the dimensions of the square that can be inscribed within the paper. The diagonal of this square will be the diameter of the circle.
Step 1: Calculate the length of the diagonal of the rectangular construction paper using the Pythagorean theorem.
- The width of the paper is 8 1/2 inches and the length is 11 inches.
- The diagonal can be calculated as: √((width^2) + (length^2))
Plugging in the numbers:
- Diagonal = √((8.5^2) + (11^2))
- Diagonal = √(72.25 + 121)
- Diagonal = √193.25
- Diagonal ≈ 13.89 inches
Step 2: Divide the diagonal length by √2 to obtain the side length of the square.
- Side length of the square = Diagonal / √2
Plugging in the numbers:
- Side length of the square = 13.89 / √2
- Side length of the square ≈ 9.84 inches
Step 3: The area of the circle can be calculated using the formula: Area = π * Radius^2.
- The radius of the circle is half the length of its diameter.
- Radius = side length of the square / 2
Plugging in the numbers:
- Radius = 9.84 / 2
- Radius ≈ 4.92 inches
Step 4: Calculate the area of the circle.
- Area = π * (Radius^2)
Plugging in the numbers:
- Area = π * (4.92^2)
- Area ≈ 76.34 square inches
Therefore, the largest circle that can be cut from a rectangular piece of construction paper measuring 8 1/2 by 11 inches has an area of approximately 76.34 square inches.