Surface Area - Mixed Shapes worksheet for calculating exact surface area of various 3D geometric figures.

To solve the problem of finding the surface area of each shape, we need to use the appropriate formulas for each geometric shape. Let's go through each shape step by step.

1) Cone

- Formula for Surface Area of a Cone:
\[
\text{Surface Area} = \pi r (r + l)
\]
where \( r \) is the radius and \( l \) is the slant height.
- Given:
- Radius (\( r \)) = 6 ft
- Slant height (\( l \)) = 16 ft
- Calculation:
\[
\text{Surface Area} = \pi \times 6 \times (6 + 16) = \pi \times 6 \times 22 = 132\pi \, \text{ft}^2
\]

2) Triangular Prism

- Formula for Surface Area of a Triangular Prism:
\[
\text{Surface Area} = \text{Base Area} \times 2 + \text{Perimeter of Base} \times \text{Height}
\]
- Given:
- Base is a right triangle with legs 10 cm and 12 cm.
- Height of the prism = 15 cm
- First, calculate the base area:
\[
\text{Base Area} = \frac{1}{2} \times 10 \times 12 = 60 \, \text{cm}^2
\]
- Next, calculate the perimeter of the base:
- Hypotenuse of the triangle:
\[
\sqrt{10^2 + 12^2} = \sqrt{100 + 144} = \sqrt{244} = 2\sqrt{61} \, \text{cm}
\]
- Perimeter of the base:
\[
10 + 12 + 2\sqrt{61} = 22 + 2\sqrt{61} \, \text{cm}
\]
- Now, calculate the surface area:
\[
\text{Surface Area} = 2 \times 60 + (22 + 2\sqrt{61}) \times 15 = 120 + 330 + 30\sqrt{61} = 450 + 30\sqrt{61} \, \text{cm}^2
\]

3) Cylinder

- Formula for Surface Area of a Cylinder:
\[
\text{Surface Area} = 2\pi r (r + h)
\]
where \( r \) is the radius and \( h \) is the height.
- Given:
- Diameter = 14 in → Radius (\( r \)) = 7 in
- Height (\( h \)) = 18 in
- Calculation:
\[
\text{Surface Area} = 2\pi \times 7 \times (7 + 18) = 2\pi \times 7 \times 25 = 350\pi \, \text{in}^2
\]

4) Cylinder

- Formula for Surface Area of a Cylinder:
\[
\text{Surface Area} = 2\pi r (r + h)
\]
- Given:
- Diameter = 18 mm → Radius (\( r \)) = 9 mm
- Height (\( h \)) = 31 mm
- Calculation:
\[
\text{Surface Area} = 2\pi \times 9 \times (9 + 31) = 2\pi \times 9 \times 40 = 720\pi \, \text{mm}^2
\]

5) Cube

- Formula for Surface Area of a Cube:
\[
\text{Surface Area} = 6s^2
\]
where \( s \) is the side length.
- Given:
- Side length (\( s \)) = 11 ft
- Calculation:
\[
\text{Surface Area} = 6 \times 11^2 = 6 \times 121 = 726 \, \text{ft}^2
\]

6) Cone

- Formula for Surface Area of a Cone:
\[
\text{Surface Area} = \pi r (r + l)
\]
- Given:
- Radius (\( r \)) = 9 m
- Slant height (\( l \)) = 17 m
- Calculation:
\[
\text{Surface Area} = \pi \times 9 \times (9 + 17) = \pi \times 9 \times 26 = 234\pi \, \text{m}^2
\]

7) Rectangular Prism

- Formula for Surface Area of a Rectangular Prism:
\[
\text{Surface Area} = 2lw + 2lh + 2wh
\]
where \( l \), \( w \), and \( h \) are the length, width, and height, respectively.
- Given:
- Length (\( l \)) = 14 cm
- Width (\( w \)) = 7 cm
- Height (\( h \)) = 9 cm
- Calculation:
\[
\text{Surface Area} = 2 \times 14 \times 7 + 2 \times 14 \times 9 + 2 \times 7 \times 9 = 196 + 252 + 126 = 574 \, \text{cm}^2
\]

8) Hemisphere

- Formula for Surface Area of a Hemisphere:
\[
\text{Surface Area} = 3\pi r^2
\]
- Given:
- Diameter = 18 cm → Radius (\( r \)) = 9 cm
- Calculation:
\[
\text{Surface Area} = 3\pi \times 9^2 = 3\pi \times 81 = 243\pi \, \text{cm}^2
\]

9) Rectangular Prism

- Formula for Surface Area of a Rectangular Prism:
\[
\text{Surface Area} = 2lw + 2lh + 2wh
\]
- Given:
- Length (\( l \)) = 7 ft
- Width (\( w \)) = 3 ft
- Height (\( h \)) = 3 ft
- Calculation:
\[
\text{Surface Area} = 2 \times 7 \times 3 + 2 \times 7 \times 3 + 2 \times 3 \times 3 = 42 + 42 + 18 = 102 \, \text{ft}^2
\]

Final Answers

\[
\boxed{
\begin{array}{ll}
1) & 132\pi \, \text{ft}^2 \\
2) & 450 + 30\sqrt{61} \, \text{cm}^2 \\
3) & 350\pi \, \text{in}^2 \\
4) & 720\pi \, \text{mm}^2 \\
5) & 726 \, \text{ft}^2 \\
6) & 234\pi \, \text{m}^2 \\
7) & 574 \, \text{cm}^2 \\
8) & 243\pi \, \text{cm}^2 \\
9) & 102 \, \text{ft}^2 \\
\end{array}
}
\]