Question

marcus designs a tabletop in the shape of a regular pentagon with a radius of 4 feet. he draws a circle around the pentagon to help him find the area of the tabletop, as shown in the diagram below.part awhat is the area, in square feet, of the circle in marcuss diagram? round your answer to the nearest hundredth.part bexplain how marcus can use the area of the circle to approximate the area of the tabletop.part cmarcus decides to change the shape of the tabletop from a regular pentagon to a regular decagon. he keeps the radius of the tabletop and the circle he draws around it the same.explain why the area of the circle is a better approximation for the area of the decagon tabletop than for the pentagon tabletop.

Explanation:

Step1: Recall circle area formula

The formula for the area of a circle is $A = \pi r^2$, where $r$ is the radius.

Step2: Substitute radius value

Substitute $r = 4$ into the formula:
$A = \pi (4)^2 = 16\pi$

Step3: Calculate and round

Compute the numerical value and round to the nearest hundredth:
$16\pi \approx 16 \times 3.1416 = 50.2656 \approx 50.27$

A regular pentagon inscribed in a circle leaves noticeable gaps between its edges and the circle's circumference. The area of the circle is larger than the pentagon's area, so it overestimates the tabletop area by the size of these gaps.

A regular dodecagon (12-sided shape) has more sides than a regular pentagon (5-sided shape). When inscribed in the same circle, a polygon with more sides fits more tightly against the circle's circumference, leaving smaller gaps between its edges and the circle. This means the circle's area is closer to the dodecagon's area than it is to the pentagon's area, so it is a better approximation.

Answer:

50.27 square feet

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