Question

find the area of the smaller sector.

area = ? \\(\text{m}^2\\)

round your answer to the nearest hundredth.

Explanation:

<pre_analysis>
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"new_concepts": [
"Sector Area Formula"
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<reasoning>

Identify the given parameters

\[

$$\begin{aligned} r &= 8\text{ m} \\ \theta &= 140^\circ \end{aligned}$$

\]

Apply the sector area formula

\[

$$\begin{aligned} \text{Area} &= \frac{\theta}{360^\circ} \cdot \pi r^2 \\ \text{Area} &= \frac{140}{360} \cdot \pi \cdot 8^2 \\ \text{Area} &= \frac{7}{18} \cdot 64\pi \\ \text{Area} &= \frac{224\pi}{9} \end{aligned}$$

\]

Calculate and round the final value

\[

$$\begin{aligned} \text{Area} &\approx \frac{224 \cdot 3.14159265}{9} \\ \text{Area} &\approx 78.19079 \\ \text{Area} &\approx 78.19\text{ m}^2 \end{aligned}$$

\]
</reasoning>

<answer>
Find the area of the smaller sector.

Area = <blank>78.19</blank> \(\text{m}^2\)

Round your answer to the nearest hundredth.
</answer>

<post_analysis>
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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Sector Area Formula"
]
}
</post_analysis>

Answer:

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"step_cot",
"blank_filling_formatter",
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],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Sector Area Formula"
],
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"Sector Area Formula"
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}
</pre_analysis>

<reasoning>

Identify the given parameters

\[

$$\begin{aligned} r &= 8\text{ m} \\ \theta &= 140^\circ \end{aligned}$$

\]

Apply the sector area formula

\[

$$\begin{aligned} \text{Area} &= \frac{\theta}{360^\circ} \cdot \pi r^2 \\ \text{Area} &= \frac{140}{360} \cdot \pi \cdot 8^2 \\ \text{Area} &= \frac{7}{18} \cdot 64\pi \\ \text{Area} &= \frac{224\pi}{9} \end{aligned}$$

\]

Calculate and round the final value

\[

$$\begin{aligned} \text{Area} &\approx \frac{224 \cdot 3.14159265}{9} \\ \text{Area} &\approx 78.19079 \\ \text{Area} &\approx 78.19\text{ m}^2 \end{aligned}$$

\]
</reasoning>

<answer>
Find the area of the smaller sector.

Area = <blank>78.19</blank> \(\text{m}^2\)

Round your answer to the nearest hundredth.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Sector Area Formula"
]
}
</post_analysis>