Question

what estimate is closest to the area of a circle with a radius of 2.5 cm?
a) 27 cm²
b) 10.5 cm²
c) 5 cm²

Explanation:

Step1: Recall the formula for the area of a circle

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

Step2: Substitute the given radius into the formula

Given \(r = 2.5\) cm. Then \(A=\pi\times(2.5)^{2}=\pi\times6.25\).

Step3: Approximate the value of \(\pi\)

Take \(\pi\approx 3.14\). So \(A\approx3.14\times6.25 = 19.625\) when \(\pi = 3.14\). If we take \(\pi\approx\frac{22}{7}\), then \(A=\frac{22}{7}\times6.25=\frac{22\times6.25}{7}=\frac{137.5}{7}\approx19.64\). Another common approximation: if we take \(\pi = 3\), then \(A = 3\times6.25=18.75\).

Answer:

A. \(27\mathrm{cm}^{2}\) (Note: There might be a miscalculation in the problem - if we assume a diameter of \(2.5\) cm (radius \(r = 1.25\) cm), \(A=\pi r^{2}\approx3.14\times(1.25)^{2}=3.14\times1.5625 = 4.90625\approx5\mathrm{cm}^{2}\) (option C). But if we follow the radius \(r = 2.5\) cm and use \(\pi\approx3.14\), the closest among the given options (assuming some error in problem - maybe diameter was intended as radius) is \(A\approx 19.6\), but if we consider wrong - input (radius vs diameter confusion), if radius \(r = 2.5\) and wrong \(\pi\approx 3.4\), \(A=\pi r^{2}\approx3.4\times6.25 = 21.25\) still not \(27\). If we consider circumference formula mis - applied as area (wrong approach \(C = 2\pi r\approx2\times3.14\times2.5=15.7\) not relevant). If we assume a typo and radius \(r = 3\) (close to \(2.5\)), \(A=\pi r^{2}\approx3.14\times9 = 28.26\approx27\)). So based on the given options and possible mis - interpretations (radius vs diameter or \(\pi\) approximation error in problem - making), the answer is A. \(27\mathrm{cm}^{2}\)