Question

(1)
(ans.)

(2)
(ans.)

(3)
(ans.)

(4)
(ans.)

(5)
(ans.)

(6)
(ans.)

(7)
(ans.)

(8)
(ans.)

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 8,
"skills_matched": [
"step_cot",
"table_formatter",
"explain_then_answer",
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],
"needs_drawing": false,
"concepts_used": [
"Area of Circle",
"Area of Semicircle",
"Area of Sector"
],
"new_concepts": [
"Area of Annulus",
"Shaded Area Decomposition"
],
"current_concepts": [
"Area of Circle",
"Area of Semicircle",
"Area of Sector",
"Area of Annulus",
"Shaded Area Decomposition"
]
}
</pre_analysis>

<reasoning>

Calculate area of question 1

Using the Area of Circle knowledge point:

• Outer radius \(R = 4\text{ cm}\)
• Inner radius \(r = 2\text{ cm}\)
• Shaded area:

\[
A_1 = \pi R^2 - \pi r^2 = \pi (4^2 - 2^2) = 12\pi\text{ cm}^2 \approx 37.70\text{ cm}^2
\]

Calculate area of question 2

Using the Area of Circle knowledge point:

• Large circle diameter \(D = 10\text{ cm} \implies R = 5\text{ cm}\)
• Small circle diameter \(d = 5\text{ cm} \implies r = 2.5\text{ cm}\)
• Shaded area:

\[
A_2 = \pi R^2 - \pi r^2 = \pi (5^2 - 2.5^2) = 18.75\pi\text{ cm}^2 \approx 58.90\text{ cm}^2
\]

Calculate area of question 3

Using the Area of Circle knowledge point:

• Square side \(s = 10\text{ cm} \implies \text{Area}_{\text{square}} = 100\text{ cm}^2\)
• Circle diameter \(d = 10\text{ cm} \implies r = 5\text{ cm} \implies \text{Area}_{\text{circle}} = 25\pi\text{ cm}^2\)
• Shaded area:

\[
A_3 = 100 - 25\pi\text{ cm}^2 \approx 21.46\text{ cm}^2
\]

Calculate area of question 4

Using the Area of Semicircle knowledge point:

• Square side \(s = 8\text{ cm} \implies \text{Area}_{\text{square}} = 64\text{ cm}^2\)
• Two semicircles of diameter \(8\text{ cm}\) combine to form one circle of radius \(r = 4\text{ cm}\)
• Shaded area:

\[
A_4 = 64 - \pi (4^2) = 64 - 16\pi\text{ cm}^2 \approx 13.73\text{ cm}^2
\]

Calculate area of question 5

Using the Area of Semicircle and Area of Circle knowledge points:

• Large semicircle diameter \(D = 6\text{ in} \implies R = 3\text{ in} \implies \text{Area}_{\text{semicircle}} = \frac{1}{2}\pi (3^2) = 4.5\pi\text{ in}^2\)
• Inner circle diameter \(d = 3\text{ in} \implies r = 1.5\text{ in} \implies \text{Area}_{\text{circle}} = \pi (1.5^2) = 2.25\pi\text{ in}^2\)
• Shaded area:

\[
A_5 = 4.5\pi - 2.25\pi = 2.25\pi\text{ in}^2 \approx 7.07\text{ in}^2
\]

Calculate area of question 6

Using the Area of Semicircle knowledge point:

• Large semicircle diameter \(D = 12\text{ in} \implies R = 6\text{ in} \implies \text{Area}_{\text{large}} = \frac{1}{2}\pi (6^2) = 18\pi\text{ in}^2\)
• Small semicircle diameter \(d = 10\text{ in} \implies r = 5\text{ in} \implies \text{Area}_{\text{small}} = \frac{1}{2}\pi (5^2) = 12.5\pi\text{ in}^2\)
• Shaded area:

\[
A_6 = 18\pi - 12.5\pi = 5.5\pi\text{ in}^2 \approx 17.28\text{ in}^2
\]

Calculate area of question 7

Using the Area of Sector and Area of Semicircle knowledge points:

• Quarter circle (sector) radius \(R = 6\text{ in} \implies \text{Area}_{\text{sector}} = \frac{1}{4}\pi (6^2) = 9\pi\text{ in}^2\)
• Semicircle diameter \(d = 6\text{ in} \implies r = 3\text{ in} \implies \text{Area}_{\text{semicircle}} = \frac{1}{2}\pi (3^2) = 4.5\pi\text{ in}^2\)
• Shaded area:

\[
A_7 = 9\pi - 4.5\pi = 4.5\pi\text{ in}^2 \approx 14.14\text{ in}^2
\]

Calculate area of question 8

Using the Area of Semicircle and Area of Sector knowledge points:

• Semicircle diameter \(d = 8\text{ in} \implies r = 4\text{ in} \implies \text{Area}_{\text{semicircle}} = \frac{1}{2}\pi (4^2) = 8\pi\text{ in}^2\)
• Quarter circle radius \(R = 4\text{ in} \implies \text{Area}_{\text{sector}} = \frac{1}{4}\pi (4^2) = 4\pi\text{ in}^2\)
• Shaded area:

\[
A_8 = 8\pi + 4\pi = 12\pi\text{ in}^2 \approx 37.70\text{ in}^2
\]
</reasoning>

<answer>

No.	Answer (Exact)	Answer (Approximate, \(\pi \approx 3.1416\))
(2)	\(18.75\pi\text{ cm}^2\)	\(58.90\text{ cm}^2\)
(3)	\(100 - 25\pi\text{ cm}^2\)	\(21.46\text{ cm}^2\)
(4)	\(64 - 16\pi\text{ cm}^2\)	\(13.73\text{ cm}^2\)
(5)	\(2.25\pi\text{ in}^2\)	\(7.07\text{ in}^2\)
(6)	\(5.5\pi\text{ in}^2\)	\(17.28\text{ in}^2\)
(7)	\(4.5\pi\text{ in}^2\)	\(14.14\text{ in}^2\)
(8)	\(12\pi\text{ in}^2\)	\(37.70\text{ in}^2\)

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Area of Circle"
]
}
</post_analysis>