Question

the circumference of a circle is 104.876 millimeters. what is the circles area? c = 104.876 mm round your answer to the nearest hundredth. square millimeters

Explanation:

Step1: Find the radius from circumference

The formula for the circumference of a circle is \( C = 2\pi r \). We know \( C = 104.876 \) mm. Solving for \( r \):
\( r=\frac{C}{2\pi}=\frac{104.876}{2\times3.1416}\) (using \(\pi\approx3.1416\))
\( r=\frac{104.876}{6.2832}\approx16.69\) mm (wait, actually, let's calculate more accurately: \( 104.876\div(2\times3.14) = 104.876\div6.28 = 16.7\) mm? Wait, no, let's do it properly. Wait, \( 2\pi r = 104.876 \), so \( r = 104.876/(2\pi) \). Let's compute \( 104.876\div(2\times3.14) = 104.876\div6.28 = 16.7 \)? Wait, 6.28×16 = 100.48, 6.28×16.7 = 6.28×16 + 6.28×0.7 = 100.48 + 4.396 = 104.876. Oh! So \( r = 16.7 \) mm.

Step2: Calculate the area

The formula for the area of a circle is \( A=\pi r^2 \). Substitute \( r = 16.7 \) mm:
\( A = 3.14\times(16.7)^2 \)
First, \( 16.7^2 = 16.7\times16.7 = 278.89 \)
Then, \( A = 3.14\times278.89 = 875.7146 \)
Wait, but let's check with the exact \( r \) from circumference. Wait, actually, when \( C = 104.876 \), \( r = C/(2\pi) = 104.876/(2\times3.14159265) \approx 104.876/6.2831853 \approx 16.699 \) mm (almost 16.7). Then \( A = \pi r^2 = 3.1416\times(16.699)^2 \approx 3.1416\times278.856 \approx 876.0 \)? Wait, no, wait, let's recalculate:

Wait, maybe I made a mistake in Step1. Let's do Step1 again:
\( C = 2\pi r \implies r = C/(2\pi) \)
Given \( C = 104.876 \), so \( r = 104.876 / (2 * 3.14) \) (using \(\pi = 3.14\) for simplicity, as maybe the problem expects that).
\( 104.876 ÷ 6.28 = 16.7 \) (exactly, because 6.28×16.7 = 6.28×16 + 6.28×0.7 = 100.48 + 4.396 = 104.876). Perfect, so \( r = 16.7 \) mm.

Step3: Compute the area

Now, area \( A = \pi r^2 = 3.14 \times (16.7)^2 \)
\( 16.7^2 = 16.7 \times 16.7 = 278.89 \)
\( A = 3.14 \times 278.89 = 875.7146 \)
Rounding to the nearest hundredth, that's 875.71 square millimeters? Wait, no, wait, 3.14×278.89: 278.89×3 = 836.67, 278.89×0.14 = 39.0446, so total 836.67 + 39.0446 = 875.7146, which is 875.71 when rounded to the nearest hundredth? Wait, but wait, maybe we should use a more accurate \(\pi\). Let's use \(\pi = 3.1416\):

\( r = 104.876 / (2×3.1416) = 104.876 / 6.2832 ≈ 16.699 \) mm (≈16.7 mm)
\( A = 3.1416×(16.699)^2 ≈ 3.1416×278.856 ≈ 876.0 \)? Wait, no, 278.856×3.1416: 278.856×3 = 836.568, 278.856×0.1416≈278.856×0.1 = 27.8856, 278.856×0.04 = 11.15424, 278.856×0.0016≈0.44617. So total 27.8856+11.15424=39.03984+0.44617≈39.486. Then total area≈836.568+39.486≈876.054, which rounds to 876.05? Wait, I think I messed up the initial radius calculation. Wait, the circumference is 104.876, which is exactly 2×3.14×16.7, so radius is 16.7. Then area is 3.14×16.7²=3.14×278.89=875.7146, which is 875.71 when rounded to the nearest hundredth. Wait, but maybe the problem expects using \(\pi=3.14\), so let's confirm:

\( r = 104.876 / (2×3.14) = 104.876 / 6.28 = 16.7 \) (exact, because 6.28×16.7=104.876). Then area is 3.14×16.7²=3.14×278.89=875.7146, which is 875.71 when rounded to the nearest hundredth. Wait, but maybe I made a mistake. Wait, 16.7 squared is 278.89, times 3.14: 278.89×3=836.67, 278.89×0.14=39.0446, sum is 836.67+39.0446=875.7146, so to the nearest hundredth, that's 875.71. But wait, let's check with a calculator: 16.7×16.7=278.89. 278.89×3.14=875.7146. Yes, so the area is approximately 875.71 square millimeters.

Wait, but maybe the circumference is given as 104.876, which is 2×3.14×16.7, so radius is 16.7. Then area is \(\pi r^2 = 3.14×16.7² = 875.71\) (rounded to nearest hundredth).

Answer:

875.71