Question

the circumference of a circle is 83.524 millimeters. what is the circles area? round your answer to the nearest hundredth. square millimeters

Explanation:

Step1: Recall circle formulas

The circumference of a circle is \( C = 2\pi r \) or \( C=\pi d \), and the area is \( A=\pi r^{2} \). First, find the radius \( r \) from the circumference \( C = 83.524 \) mm. Using \( C = 2\pi r \), solve for \( r \): \( r=\frac{C}{2\pi} \).

Step2: Calculate the radius

Substitute \( C = 83.524 \) and \( \pi\approx3.1416 \) into the formula: \( r=\frac{83.524}{2\times3.1416}=\frac{83.524}{6.2832}\approx13.3 \) mm (we can also keep more decimals for accuracy, but let's do it precisely: \( 83.524\div6.2832 = 13.3 \) (exact here as \( 6.2832\times13.3 = 83.56656 \), wait, maybe better to use \( \pi = 3.14 \) for simplicity? Wait, let's recalculate. If \( C = 2\pi r \), then \( r=\frac{C}{2\pi} \). Let's use \( \pi\approx3.14 \): \( r=\frac{83.524}{2\times3.14}=\frac{83.524}{6.28}=13.3 \) mm (since \( 6.28\times13.3 = 83.524 \), perfect!).

Step3: Calculate the area

Now use the area formula \( A=\pi r^{2} \). Substitute \( r = 13.3 \) and \( \pi\approx3.14 \): \( A = 3.14\times(13.3)^{2} \). First, calculate \( 13.3^{2}=13.3\times13.3 = 176.89 \). Then \( A = 3.14\times176.89 = 555.4346 \). Round to the nearest hundredth: \( 555.43 \) (wait, no, 3.14×176.89: 176.89×3 = 530.67, 176.89×0.14 = 24.7646, total 530.67 + 24.7646 = 555.4346, which rounds to 555.43? Wait, but maybe we should use a more accurate \( \pi \). Let's use \( \pi = 3.1416 \). Then \( r=\frac{83.524}{2\times3.1416}=\frac{83.524}{6.2832}\approx13.293 \) mm. Then \( A=\pi r^{2}=3.1416\times(13.293)^{2} \). Calculate \( 13.293^{2}=13.293\times13.293\approx176.7 \) (wait, no, 13.293×13.293: 13×13 = 169, 13×0.293 = 3.809, 0.293×13 = 3.809, 0.293×0.293≈0.0858, so (13 + 0.293)² = 13² + 2×13×0.293 + 0.293² = 169 + 7.618 + 0.0858≈176.7038. Then \( A = 3.1416\times176.7038\approx3.1416\times176.7038 \). Let's calculate 176.7038×3 = 530.1114, 176.7038×0.1416≈176.7038×0.1 = 17.67038, 176.7038×0.04 = 7.06815, 176.7038×0.0016≈0.2827. So total 17.67038 + 7.06815 = 24.73853 + 0.2827≈25.02123. Then total area≈530.1114 + 25.02123≈555.1326. Wait, there's a discrepancy because of \( r \) calculation. Wait, the exact circumference \( C = 83.524 \), so \( r=\frac{83.524}{2\pi} \). Let's compute \( 83.524\div(2\times3.1415926535)=83.524\div6.283185307≈13.293 \) (as before). Then \( r^{2}=13.293^{2}=176.702 \) (more accurately, 13.293×13.293: 13.293×13 = 172.809, 13.293×0.293 = 3.894849, so total 172.809 + 3.894849 = 176.703849). Then \( A=\pi r^{2}=3.1415926535\times176.703849≈3.1415926535×176.703849 \). Let's do this multiplication: 176.703849×3 = 530.111547, 176.703849×0.1415926535≈176.703849×0.1 = 17.6703849, 176.703849×0.04 = 7.06815396, 176.703849×0.0015926535≈0.2814. So 17.6703849 + 7.06815396 = 24.73853886 + 0.2814≈25.0199. Then total area≈530.111547 + 25.0199≈555.1314. Wait, but the initial circumference is 83.524, which is very close to \( 2\pi\times13.3 \) (since \( 2\times3.14\times13.3 = 83.524 \), exactly). So if we use \( r = 13.3 \), then \( A=\pi\times13.3^{2}=3.14\times176.89 = 555.4346 \), which rounds to 555.43? But there's a mistake here: 2×3.14×13.3 = 83.524, exactly. So \( r = 13.3 \) is exact for \( \pi = 3.14 \). So then \( A = 3.14\times13.3^{2} \). Calculate \( 13.3^{2}=176.89 \), then \( 3.14\times176.89 = 555.4346 \). Rounding to the nearest hundredth: look at the thousandth place, which is 4, so we round down? Wait, 555.4346: the hundredth place is 3, thousandth is 4, so it's 555.43? Wait, no: 555.4346, the digits are: 5 (hundreds), 5 (tens), 5 (units),. (decimal), 4 (tenths), 3 (hundredths), 4 (thousandths), 6 (ten - thousandths). To round to the nearest hundredth, we look at the thousandth digit, which is 4. Since 4 < 5, we keep the hundredth digit as is. So 555.43. But wait, if we use a more accurate \( \pi \), we get a different result, but since the circumference is given as 83.524, which is a multiple of \( 2\times3.14 \) (since \( 2\times3.14 = 6.28 \), and \( 6.28\times13.3 = 83.524 \)), so we should use \( \pi = 3.14 \) here for consistency. So the area is \( 3.14\times13.3^{2}=555.4346\approx555.43 \) square millimeters. Wait, but let's check again: \( 13.3\times13.3 = 176.89 \), \( 176.89\times3.14 \): 176.89×3 = 530.67, 176.89×0.14 = 24.7646, sum is 530.67 + 24.7646 = 555.4346. Yes, so rounding to the nearest hundredth is 555.43.

Answer:

\( 555.43 \) square millimeters