Math worksheet for calculating volume and surface area of cones and spheres.

Let's solve each problem step by step using the formulas for volume and surface area of spheres and cones.

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🔷 Formulas:

#### 🟨 Sphere:
- Volume: $ V = \frac{4}{3} \pi r^3 $
- Surface Area: $ A = 4\pi r^2 $

#### 🟦 Cone:
- Volume: $ V = \frac{1}{3} \pi r^2 h $
- Curved Surface Area (CSA): $ A = \pi r l $, where $ l = \sqrt{r^2 + h^2} $
- Total Surface Area (TSA): $ A = \pi r l + \pi r^2 $

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We'll go through each question.

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1) Sphere with radius = 2 cm

- Volume:
$$
V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \approx 33.51 \text{ cm}^3
$$

- Surface Area:
$$
A = 4\pi r^2 = 4\pi (2)^2 = 4\pi (4) = 16\pi \approx 50.27 \text{ cm}^2
$$

✔ Answer:
- Volume = 33.51 cm³
- Surface Area = 50.27 cm²

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2) Sphere with diameter = 14 mm → radius = 7 mm

- Volume:
$$
V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (7)^3 = \frac{4}{3} \pi (343) = \frac{1372}{3} \pi \approx 1436.76 \text{ mm}^3
$$

- Surface Area:
$$
A = 4\pi r^2 = 4\pi (7)^2 = 4\pi (49) = 196\pi \approx 615.75 \text{ mm}^2
$$

✔ Answer:
- Volume = 1436.76 mm³
- Surface Area = 615.75 mm²

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3) Sphere with volume = 180 cm³ → find radius

Use:
$$
V = \frac{4}{3} \pi r^3 = 180
$$
Solve for $ r $:

$$
r^3 = \frac{180 \times 3}{4\pi} = \frac{540}{4\pi} = \frac{135}{\pi}
$$
$$
r = \sqrt[3]{\frac{135}{\pi}} \approx \sqrt[3]{42.97} \approx 3.50 \text{ cm}
$$

✔ Answer:
- Radius ≈ 3.50 cm

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4) Sphere with surface area = 25 mm² → find radius

Use:
$$
A = 4\pi r^2 = 25
$$
$$
r^2 = \frac{25}{4\pi} \approx \frac{25}{12.566} \approx 1.989
$$
$$
r = \sqrt{1.989} \approx 1.41 \text{ mm}
$$

✔ Answer:
- Radius ≈ 1.41 mm

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5) Cone: r = 5 cm, h = 12 cm, slant height l = 13 cm

Given: $ r = 5 $, $ h = 12 $, $ l = 13 $

- Curved Surface Area (CSA):
$$
CSA = \pi r l = \pi \cdot 5 \cdot 13 = 65\pi \approx 204.20 \text{ cm}^2
$$

- Total Surface Area (TSA):
$$
TSA = \pi r l + \pi r^2 = 65\pi + \pi (25) = 90\pi \approx 282.74 \text{ cm}^2
$$

✔ Answer:
- Curved surface area = 204.20 cm²
- Total surface area = 282.74 cm²

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6) Cone: r = 7 cm, slant height l = 25 cm

We are not given height, but we can use $ l = 25 $, $ r = 7 $

- Volume: Need height $ h $. Use Pythagoras:
$$
h = \sqrt{l^2 - r^2} = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24 \text{ cm}
$$

Now compute volume:
$$
V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (49)(24) = \frac{1}{3} \pi (1176) = 392\pi \approx 1231.51 \text{ cm}^3
$$

- Total Surface Area:
$$
TSA = \pi r l + \pi r^2 = \pi(7)(25) + \pi(49) = 175\pi + 49\pi = 224\pi \approx 703.72 \text{ cm}^2
$$

✔ Answer:
- Volume = 1231.51 cm³
- Total surface area = 703.72 cm²

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7) Cone: diameter = 14.4 cm → r = 7.2 cm, h = 9.6 cm

First find slant height $ l $:
$$
l = \sqrt{r^2 + h^2} = \sqrt{(7.2)^2 + (9.6)^2} = \sqrt{51.84 + 92.16} = \sqrt{144} = 12 \text{ cm}
$$

- Volume:
$$
V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (7.2)^2 (9.6) = \frac{1}{3} \pi (51.84)(9.6)
$$
$$
= \frac{1}{3} \pi (500.352) = 166.784\pi \approx 523.84 \text{ cm}^3
$$

- Total Surface Area:
$$
TSA = \pi r l + \pi r^2 = \pi(7.2)(12) + \pi(51.84) = 86.4\pi + 51.84\pi = 138.24\pi \approx 434.29 \text{ cm}^2
$$

✔ Answer:
- Volume = 523.84 cm³
- Total surface area = 434.29 cm²

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8) Composite shape: Cone + Hemisphere

- Cone: height = 23 cm, hemisphere radius = 15 cm
So cone base radius = 15 cm
Cone height = 23 cm
Hemisphere is attached to the base of the cone.

We need:
- Volume = Volume of cone + Volume of hemisphere
- Total surface area = CSA of cone + CSA of hemisphere (since base of hemisphere is glued to cone, no base area)

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#### 🔹 Volume:

- Cone volume:
$$
V_{\text{cone}} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (15)^2 (23) = \frac{1}{3} \pi (225)(23) = \frac{1}{3} \pi (5175) = 1725\pi
$$

- Hemisphere volume:
$$
V_{\text{hemi}} = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi (3375) = 2250\pi
$$

- Total volume:
$$
V = 1725\pi + 2250\pi = 3975\pi \approx 12474.37 \text{ cm}^3
$$

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#### 🔹 Total Surface Area:

- Curved surface area of cone:
First find slant height $ l $:
$$
l = \sqrt{r^2 + h^2} = \sqrt{15^2 + 23^2} = \sqrt{225 + 529} = \sqrt{754} \approx 27.46 \text{ cm}
$$
$$
CSA_{\text{cone}} = \pi r l = \pi (15)(27.46) \approx 411.9\pi \approx 1293.37 \text{ cm}^2
$$

- Curved surface area of hemisphere:
$$
CSA_{\text{hemi}} = 2\pi r^2 = 2\pi (225) = 450\pi \approx 1413.72 \text{ cm}^2
$$

- Total surface area (no base since they're joined):
$$
TSA = 1293.37 + 1413.72 = 2707.09 \text{ cm}^2
$$

✔ Answer:
- Volume = 12474.37 cm³
- Total area = 2707.09 cm²

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✔ Final Answers Summary:

| Q | Answer |
|---|--------|
| 1) | Volume = 33.51 cm³, Surface Area = 50.27 cm² |
| 2) | Volume = 1436.76 mm³, Surface Area = 615.75 mm² |
| 3) | Radius = 3.50 cm |
| 4) | Radius = 1.41 mm |
| 5) | Curved SA = 204.20 cm², Total SA = 282.74 cm² |
| 6) | Volume = 1231.51 cm³, Total SA = 703.72 cm² |
| 7) | Volume = 523.84 cm³, Total SA = 434.29 cm² |
| 8) | Volume = 12474.37 cm³, Total Area = 2707.09 cm² |

> All answers rounded to 2 decimal places as required.

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