Student's completed homework assignment showing calculations for the circumference of circles and the perimeter of various polygons.

It looks like you've uploaded a worksheet titled "Circumference of Circles - Geometry", which contains several problems involving calculating the circumference and area of circles, as well as some composite shapes. The worksheet has been filled out with handwritten solutions in blue ink.

Let me go through each problem systematically and explain the solutions:

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Problem 1: Circle with radius = 8 cm

- Given: Radius $ r = 8 $ cm
- Circumference formula: $ C = 2\pi r $
- Calculation:
$$
C = 2 \times \pi \times 8 = 16\pi \approx 50.24 \text{ cm}
$$
- Area formula: $ A = \pi r^2 $
- Calculation:
$$
A = \pi \times 8^2 = 64\pi \approx 200.96 \text{ cm}^2
$$

✔ Correct — Answers are $ 50.24 $ cm and $ 200.96 $ cm² (using $ \pi \approx 3.14 $).

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Problem 2: Rectangle with semicircle on top

- Rectangle dimensions: 12 ft × 5 ft
- Semicircle diameter: 12 ft → radius $ r = 6 $ ft

#### Perimeter:
- Rectangle sides: two vertical sides = $ 2 \times 5 = 10 $ ft
- Bottom side: 12 ft
- Top is a semicircle, so arc length = $ \frac{1}{2} \times 2\pi r = \pi r = \pi \times 6 = 18.84 $ ft
- Total perimeter = $ 12 + 10 + 18.84 = 40.84 $ ft

#### Area:
- Rectangle area: $ 12 \times 5 = 60 $ ft²
- Semicircle area: $ \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (6)^2 = 18\pi \approx 56.52 $ ft²
- Total area: $ 60 + 56.52 = 116.52 $ ft²

✔ Correct — Perimeter ≈ 40.84 ft, Area ≈ 116.52 ft²

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Problem 3: Circle with diameter = 14 m

- Diameter $ d = 14 $ m → radius $ r = 7 $ m
- Circumference: $ C = \pi d = 3.14 \times 14 = 43.96 $ m
- Area: $ A = \pi r^2 = 3.14 \times 49 = 153.86 $ m²

✔ Correct

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Problem 4: Circle with radius = 6.5 cm

- $ r = 6.5 $ cm
- Circumference: $ 2\pi r = 2 \times 3.14 \times 6.5 = 40.82 $ cm
- Area: $ \pi r^2 = 3.14 \times (6.5)^2 = 3.14 \times 42.25 = 132.665 $ cm² ≈ 132.67 cm²

✔ Correct

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Problem 5: Square with quarter circle cut out

- Square side = 10 cm
- Quarter circle radius = 10 cm (same as side)

#### Perimeter:
- Three sides of square: $ 3 \times 10 = 30 $ cm
- One curved edge: quarter circumference = $ \frac{1}{4} \times 2\pi r = \frac{1}{2} \pi r = \frac{1}{2} \times 3.14 \times 10 = 15.7 $ cm
- Total perimeter = $ 30 + 15.7 = 45.7 $ cm

#### Area:
- Square area: $ 10 \times 10 = 100 $ cm²
- Quarter circle area: $ \frac{1}{4} \pi r^2 = \frac{1}{4} \times 3.14 \times 100 = 78.5 $ cm²
- Remaining area: $ 100 - 78.5 = 21.5 $ cm²

⚠️ Error here!
- The student wrote "Area = 100 - 78.5 = 21.5", but this is incorrect.
- The shape shown is a square with a quarter circle removed, so yes, subtracting is correct.
- But wait: the figure appears to be a square with a quarter circle added? Let's check the diagram.

Actually, based on the drawing and label "quarter circle", it seems like the quarter circle is cut out from the square. So subtraction is correct.

But let’s double-check the perimeter:
- The three straight sides: 3 × 10 = 30
- The curved part: quarter of full circumference: $ \frac{1}{4} \times 2\pi r = \frac{1}{2} \pi r = 15.7 $
- So total perimeter = 30 + 15.7 = 45.7 cm ✔

But the area should be:
$$
A = \text{Square} - \text{Quarter circle} = 100 - 78.5 = 21.5 \text{ cm}^2
$$
So that is correct.

Wait — actually, looking at the image, the student wrote 100 - 78.5 = 21.5, which is correct.

✔ Correct

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Problem 6: Rectangle with semicircle on one end

- Rectangle: 12 cm × 6 cm
- Semicircle diameter = 12 cm → radius = 6 cm

#### Perimeter:
- Two vertical sides: $ 2 \times 6 = 12 $ cm
- One horizontal side: 12 cm (bottom)
- Semicircular arc: $ \pi r = 3.14 \times 6 = 18.84 $ cm
- Top side: not included (replaced by semicircle)
- Total perimeter = $ 12 + 12 + 18.84 = 42.84 $ cm

#### Area:
- Rectangle: $ 12 \times 6 = 72 $ cm²
- Semicircle: $ \frac{1}{2} \pi r^2 = \frac{1}{2} \times 3.14 \times 36 = 56.52 $ cm²
- Total area: $ 72 + 56.52 = 128.52 $ cm²

✔ Correct

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Problem 7: Circle with radius = 5.5 cm

- $ r = 5.5 $
- $ C = 2\pi r = 2 \times 3.14 \times 5.5 = 34.54 $ cm
- $ A = \pi r^2 = 3.14 \times 30.25 = 94.985 $ cm² ≈ 94.99 cm²

✔ Correct

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Problem 8: Circle with diameter = 12 in

- $ d = 12 $ → $ r = 6 $
- $ C = \pi d = 3.14 \times 12 = 37.68 $ in
- $ A = \pi r^2 = 3.14 \times 36 = 113.04 $ in²

✔ Correct

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✔ Summary of All Problems:

All answers are correctly calculated, using $ \pi \approx 3.14 $. The student applied formulas properly for:
- Circumference: $ C = 2\pi r $ or $ C = \pi d $
- Area: $ A = \pi r^2 $
- Composite shapes: combining rectangles and circular parts

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🔍 Final Notes:

- For composite shapes, always:
- Identify all components (rectangle, semicircle, etc.)
- Add or subtract areas appropriately
- For perimeter, only include outer edges (no internal lines)

This worksheet demonstrates good understanding of geometry concepts related to circles and their applications in real-world shapes.

Let me know if you'd like a PDF version of the corrected worksheet or want explanations in another format!