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The circle shown has center $F$ and radius 6
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29 Aug 2025, 00:19
29 Aug 2025, 00:19
Expert Reply
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The circle shown has center $F$ and radius 6 . If the measure of angle $\(E F G\)$ is $\(60^{\circ}\)$, what is the combined area of the shaded regions, rounded to the nearest integer?
A. \(\sqrt{2}\)
B. 36
C. 37
D. 38
E. 40
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The circle shown has center $F$ and radius 6
[#permalink]
31 Aug 2025, 08:40
31 Aug 2025, 08:40
Expert Reply
Given:
- Circle with center $F$ and radius $r=6$.
- Central angle $\(\angle E F G=60^{\circ}\)$.
- Two shaded sectors, each corresponding to the $\(60^{\circ}\)$ angle.
Step 1: Calculate the area of one sector with central angle $\(60^{\circ}\)$
Area of a sector is
$$
\(\text { Area of sector }=\frac{\theta}{360^{\circ}} \times \pi r^2\)
$$
With $\(\theta=60^{\circ}\)$ and $r=6$,
$$
\(=\frac{60}{360} \times \pi \times 6^2=\frac{1}{6} \times \pi \times 36=6 \pi\)
$$
Step 2: Total shaded area (two sectors)
Since there are two identical shaded sectors,
$$
\(\text { Total shaded area }=2 \times 6 \pi=12 \pi\)
$$
Step 3: Approximate value
$$
\(12 \pi \approx 12 \times 3.1416=37.699\)
$$
Rounded to the nearest integer:
38
Final Answer:
D. 38
- Circle with center $F$ and radius $r=6$.
- Central angle $\(\angle E F G=60^{\circ}\)$.
- Two shaded sectors, each corresponding to the $\(60^{\circ}\)$ angle.
Step 1: Calculate the area of one sector with central angle $\(60^{\circ}\)$
Area of a sector is
$$
\(\text { Area of sector }=\frac{\theta}{360^{\circ}} \times \pi r^2\)
$$
With $\(\theta=60^{\circ}\)$ and $r=6$,
$$
\(=\frac{60}{360} \times \pi \times 6^2=\frac{1}{6} \times \pi \times 36=6 \pi\)
$$
Step 2: Total shaded area (two sectors)
Since there are two identical shaded sectors,
$$
\(\text { Total shaded area }=2 \times 6 \pi=12 \pi\)
$$
Step 3: Approximate value
$$
\(12 \pi \approx 12 \times 3.1416=37.699\)
$$
Rounded to the nearest integer:
38
Final Answer:
D. 38
gmatclubot
The circle shown has center $F$ and radius 6 [#permalink]
31 Aug 2025, 08:40
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