Surface area calculation worksheet for mixed geometric shapes.

Let's solve each of these surface area problems step by step. We'll use the formulas for the surface areas of common 3D shapes and round answers to two decimal places using π ≈ 3.14.

---

1) Sphere

- Diameter = 20.5 cm → Radius $ r = \frac{20.5}{2} = 10.25 $ cm
- Surface Area of a sphere:
$$
SA = 4\pi r^2 = 4 \times 3.14 \times (10.25)^2
$$
$$
= 4 \times 3.14 \times 105.0625 = 1319.47 \text{ cm}^2
$$

✔ Surface Area = 1319.47 cm²

---

2) Triangular Prism

This is a triangular prism with:
- Base triangle: base = 10.8 yd, height = 12.4 yd
- Length of prism = 14.6 yd

We need:
- Area of two triangular bases
- Area of three rectangular sides

#### Step 1: Area of one triangle
$$
A_{\text{triangle}} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 10.8 \times 12.4 = 66.96 \text{ yd}^2
$$
Two triangles: $ 2 \times 66.96 = 133.92 $

#### Step 2: Areas of rectangular faces
The three rectangles have widths equal to the sides of the triangle and length = 14.6 yd.

We know:
- One side = 10.8 yd
- One side = 12.4 yd
- The third side (hypotenuse) can be found using Pythagoras:

$$
\text{Hypotenuse} = \sqrt{10.8^2 + 12.4^2} = \sqrt{116.64 + 153.76} = \sqrt{270.4} ≈ 16.44 \text{ yd}
$$

Now compute rectangle areas:
- $ 10.8 \times 14.6 = 157.68 $
- $ 12.4 \times 14.6 = 180.24 $
- $ 16.44 \times 14.6 ≈ 239.02 $

Total rectangle area = $ 157.68 + 180.24 + 239.02 = 576.94 $

#### Total Surface Area:
$$
SA = 133.92 + 576.94 = 710.86 \text{ yd}^2
$$

✔ Surface Area = 710.86 yd²

---

3) Cube

- Side length = 5.4 ft
- Surface area of cube: $ 6s^2 $
$$
SA = 6 \times (5.4)^2 = 6 \times 29.16 = 174.96 \text{ ft}^2
$$

✔ Surface Area = 174.96 ft²

---

4) Cone

- Radius $ r = 2.5 $ m
- Slant height $ l = 12.4 $ m
- Surface Area of cone:
$$
SA = \pi r^2 + \pi r l = \pi r(r + l)
$$
$$
= 3.14 \times 2.5 \times (2.5 + 12.4) = 3.14 \times 2.5 \times 14.9
$$
$$
= 3.14 \times 37.25 = 116.87 \text{ m}^2
$$

✔ Surface Area = 116.87 m²

---

5) Rectangular Prism (Cuboid)

- Dimensions: 5.5 in × 5.5 in × 5.5 in? Wait — looks like it’s 5.5 in × 5.5 in × 5.5 in, but that would be a cube. But wait — the drawing shows different dimensions?

Wait — looking at image: It's labeled as 5.5 in on two sides and 13.5 in on the other? Let me recheck.

Actually, from the diagram:
- Height = 13.5 in
- Width = 5.5 in
- Depth = 5.5 in

So it’s a rectangular prism with:
- Two faces: $ 5.5 \times 5.5 $
- Two faces: $ 5.5 \times 13.5 $
- Two faces: $ 5.5 \times 13.5 $

Wait — actually:
- Front/back: $ 5.5 \times 13.5 $
- Left/right: $ 5.5 \times 13.5 $? No — depth is 5.5, so left/right are $ 5.5 \times 5.5 $? Let's clarify.

Standard labeling:
- Length = 13.5 in
- Width = 5.5 in
- Height = 5.5 in

Then:
- Top/Bottom: $ 13.5 \times 5.5 = 74.25 $ → two faces: $ 2 \times 74.25 = 148.5 $
- Front/Back: $ 13.5 \times 5.5 = 74.25 $ → same as above → $ 148.5 $
- Sides: $ 5.5 \times 5.5 = 30.25 $ → two sides → $ 60.5 $

Wait — no! The vertical dimension is 13.5 in, and horizontal is 5.5 in, and depth is 5.5 in.

So:
- Top/Bottom: $ 13.5 \times 5.5 = 74.25 $ → 2 × 74.25 = 148.5
- Front/Back: $ 13.5 \times 5.5 = 74.25 $ → same → 148.5
- Left/Right: $ 5.5 \times 5.5 = 30.25 $ → 2 × 30.25 = 60.5

But that can’t be — if both width and depth are 5.5, then yes.

Wait — perhaps it's a rectangular prism with:
- L = 13.5 in
- W = 5.5 in
- H = 5.5 in

Then:
- SA = 2(LW + LH + WH)
$$
= 2(13.5 \times 5.5 + 13.5 \times 5.5 + 5.5 \times 5.5)
= 2(74.25 + 74.25 + 30.25)
= 2(178.75) = 357.50 \text{ in}^2
$$

✔ Surface Area = 357.50 in²

---

6) Cylinder

- Diameter = 11.4 cm → radius $ r = 5.7 $ cm
- Height = 12.4 cm

Surface Area:
$$
SA = 2\pi r^2 + 2\pi r h = 2\pi r(r + h)
$$
$$
= 2 \times 3.14 \times 5.7 \times (5.7 + 12.4) = 2 \times 3.14 \times 5.7 \times 18.1
$$
First: $ 2 \times 3.14 = 6.28 $
$ 6.28 \times 5.7 = 35.80 $
$ 35.80 \times 18.1 ≈ 647.18 $

Let’s compute step-by-step:
- $ r^2 = 5.7^2 = 32.49 $
- $ 2\pi r^2 = 2 \times 3.14 \times 32.49 = 6.28 \times 32.49 ≈ 204.18 $
- $ 2\pi r h = 2 \times 3.14 \times 5.7 \times 12.4 = 6.28 \times 5.7 \times 12.4 $
- $ 6.28 \times 5.7 = 35.80 $
- $ 35.80 \times 12.4 ≈ 444.32 $
- Total: $ 204.18 + 444.32 = 648.50 \text{ cm}^2 $

✔ Surface Area = 648.50 cm²

---

7) Rectangular Prism

- Dimensions: 17.9 in × 7 in × 3.5 in

Use formula:
$$
SA = 2(lw + lh + wh)
$$
$$
= 2(17.9 \times 7 + 17.9 \times 3.5 + 7 \times 3.5)
= 2(125.3 + 62.65 + 24.5)
= 2(212.45) = 424.90 \text{ in}^2
$$

✔ Surface Area = 424.90 in²

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8) Hemisphere

- Radius = 5.8 in
- Surface Area of a hemisphere includes:
- Curved surface: $ 2\pi r^2 $
- Flat circular base: $ \pi r^2 $
- So total: $ 3\pi r^2 $

$$
SA = 3 \times 3.14 \times (5.8)^2 = 3 \times 3.14 \times 33.64 = 9.42 \times 33.64
$$
$$
= 317.23 \text{ in}^2
$$

✔ Surface Area = 317.23 in²

---

9) Square Pyramid

- Base: square with side = 5.4 yd
- Slant height = 11.4 yd (from apex to midpoint of base edge)

Surface Area = Base area + 4 × (area of triangular face)

- Base area = $ 5.4 \times 5.4 = 29.16 $
- Each triangle: $ \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 5.4 \times 11.4 = 30.78 $
- Four triangles: $ 4 \times 30.78 = 123.12 $
- Total SA = $ 29.16 + 123.12 = 152.28 \text{ yd}^2 $

✔ Surface Area = 152.28 yd²

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✔ Final Answers:

| Problem | Surface Area |
|--------|--------------|
| 1) Sphere | 1319.47 cm² |
| 2) Triangular Prism | 710.86 yd² |
| 3) Cube | 174.96 ft² |
| 4) Cone | 116.87 m² |
| 5) Rectangular Prism | 357.50 in² |
| 6) Cylinder | 648.50 cm² |
| 7) Rectangular Prism | 424.90 in² |
| 8) Hemisphere | 317.23 in² |
| 9) Square Pyramid | 152.28 yd² |

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Let me know if you'd like these formatted for printing or want explanations for any specific shape!