Which of the following is the area of the largest circles that can fit entirely inside a rectangle that measures 8 centimeters by 10 centimeters?
A.
18π cm2
B.
10 π cm2
C.
16 π cm2
D.
8 π cm2
The Correct Answer and Explanation is:
To determine the area of the largest circles that can fit entirely inside a rectangle measuring 8 centimeters by 10 centimeters, follow these steps:
- Identify the Diameter of the Largest Circle: The largest circle that can fit inside the rectangle will have its diameter equal to the smaller dimension of the rectangle. This ensures that the circle fits completely within the rectangle without overlapping its boundaries. In this case, the dimensions of the rectangle are 8 cm and 10 cm. Thus, the diameter of the largest circle is the smaller of these two dimensions, which is 8 cm.
- Calculate the Radius: The radius of a circle is half of its diameter. Therefore, the radius rrr of the largest circle is:r=Diameter2=8 cm2=4 cmr = \frac{\text{Diameter}}{2} = \frac{8 \text{ cm}}{2} = 4 \text{ cm}r=2Diameter=28 cm=4 cm
- Calculate the Area of the Circle: The area AAA of a circle is given by the formula:A=πr2A = \pi r^2A=πr2Substituting the radius r=4 cmr = 4 \text{ cm}r=4 cm:A=π(4 cm)2=π×16 cm2=16π cm2A = \pi (4 \text{ cm})^2 = \pi \times 16 \text{ cm}^2 = 16 \pi \text{ cm}^2A=π(4 cm)2=π×16 cm2=16π cm2
Therefore, the area of the largest circle that can fit entirely inside the rectangle is 16π cm216 \pi \text{ cm}^216π cm2.
Answer: C. 16 π cm²
Explanation:
To fit a circle inside a rectangle, the diameter of the circle must be no greater than the smallest dimension of the rectangle. This constraint ensures that the circle will not extend beyond the edges of the rectangle. Here, the dimensions of the rectangle are 8 cm and 10 cm. The smaller dimension (8 cm) determines the maximum diameter of the circle that can fit inside the rectangle. By calculating the radius (half of the diameter), you find that the radius is 4 cm. Using the formula for the area of a circle, you substitute the radius into the formula to find the area. Thus, the area of the circle that fits perfectly within the rectangle is 16π cm216 \pi \text{ cm}^216π cm2.