Question

the radius of a quarter circle is 3 millimeters. what is the quarter circles area?
round your answer to the nearest hundredth.
square millimeters

Explanation:

Step1: Recall the area formula for a full circle

The area of a full circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle.

Step2: Determine the fraction of the circle for a quarter circle

A quarter circle is \( \frac{1}{4} \) of a full circle. So, the area of a quarter circle will be \( \frac{1}{4} \) of the area of the full circle.

Step3: Substitute the radius into the formula for the quarter - circle area

Given that the radius \( r = 3 \) mm. First, find the area of the full circle: \( A_{full}=\pi r^{2}=\pi\times(3)^{2}=9\pi \) square millimeters. Then, the area of the quarter circle \( A_{quarter}=\frac{1}{4}\times A_{full}=\frac{1}{4}\times9\pi=\frac{9\pi}{4} \) square millimeters.

Step4: Calculate the numerical value and round

We know that \( \pi\approx3.14159 \). So, \( A_{quarter}=\frac{9\times3.14159}{4}=\frac{28.27431}{4} = 7.0685775 \) square millimeters. Rounding to the nearest hundredth, we look at the thousandth place. The digit in the thousandth place is 8, which is greater than 5, so we round up the hundredth place. \( 7.0685775\approx7.07 \) square millimeters.

Answer:

\( 7.07 \) square millimeters