Volume and Surface Area of 3D Shapes worksheet - Free Printable

To solve the problem, we need to calculate the volume and surface area for each of the 3D shapes provided in the image. Let's go through each shape step by step.

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1. Rectangular Prism

#### Dimensions:
- Length (\( l \)) = 10 mm
- Width (\( w \)) = 6 mm
- Height (\( h \)) = 3 mm

#### Volume:
\[ \text{Volume} = l \times w \times h \]
\[ \text{Volume} = 10 \, \text{mm} \times 6 \, \text{mm} \times 3 \, \text{mm} = 180 \, \text{mm}^3 \]

#### Surface Area:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
\[ \text{Surface Area} = 2(10 \times 6 + 10 \times 3 + 6 \times 3) \]
\[ \text{Surface Area} = 2(60 + 30 + 18) \]
\[ \text{Surface Area} = 2 \times 108 = 216 \, \text{mm}^2 \]

Results:
- Volume = \( 180 \, \text{mm}^3 \)
- Surface Area = \( 216 \, \text{mm}^2 \)

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2. Cube

#### Dimensions:
- Side length (\( s \)) = 4 mm

#### Volume:
\[ \text{Volume} = s^3 \]
\[ \text{Volume} = 4^3 = 64 \, \text{mm}^3 \]

#### Surface Area:
\[ \text{Surface Area} = 6s^2 \]
\[ \text{Surface Area} = 6 \times 4^2 \]
\[ \text{Surface Area} = 6 \times 16 = 96 \, \text{mm}^2 \]

Results:
- Volume = \( 64 \, \text{mm}^3 \)
- Surface Area = \( 96 \, \text{mm}^2 \)

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3. Cylinder

#### Dimensions:
- Radius (\( r \)) = 7 cm
- Height (\( h \)) = 10 cm

#### Volume:
\[ \text{Volume} = \pi r^2 h \]
\[ \text{Volume} = \pi \times 7^2 \times 10 \]
\[ \text{Volume} = \pi \times 49 \times 10 \]
\[ \text{Volume} = 490\pi \, \text{cm}^3 \]
\[ \text{Volume} \approx 1539.4 \, \text{cm}^3 \]

#### Surface Area:
\[ \text{Surface Area} = 2\pi r(h + r) \]
\[ \text{Surface Area} = 2\pi \times 7 \times (10 + 7) \]
\[ \text{Surface Area} = 2\pi \times 7 \times 17 \]
\[ \text{Surface Area} = 238\pi \, \text{cm}^2 \]
\[ \text{Surface Area} \approx 748.7 \, \text{cm}^2 \]

Results:
- Volume = \( 490\pi \, \text{cm}^3 \) or \( \approx 1539.4 \, \text{cm}^3 \)
- Surface Area = \( 238\pi \, \text{cm}^2 \) or \( \approx 748.7 \, \text{cm}^2 \)

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4. Triangular Prism

#### Dimensions:
- Base triangle sides: 15 cm, 17 cm, 10 cm
- Height of prism (\( h \)) = 8 cm
- Height of the triangular base (\( h_{\text{triangle}} \)) can be calculated using Heron's formula.

##### Step 1: Calculate the semi-perimeter (\( s \)) of the triangular base:
\[ s = \frac{15 + 17 + 10}{2} = 21 \, \text{cm} \]

##### Step 2: Calculate the area of the triangular base using Heron's formula:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
\[ \text{Area} = \sqrt{21(21-15)(21-17)(21-10)} \]
\[ \text{Area} = \sqrt{21 \times 6 \times 4 \times 11} \]
\[ \text{Area} = \sqrt{5544} \]
\[ \text{Area} \approx 74.46 \, \text{cm}^2 \]

##### Step 3: Calculate the volume:
\[ \text{Volume} = \text{Base Area} \times \text{Height of Prism} \]
\[ \text{Volume} = 74.46 \times 8 \]
\[ \text{Volume} \approx 595.68 \, \text{cm}^3 \]

##### Step 4: Calculate the surface area:
\[ \text{Surface Area} = 2 \times \text{Base Area} + \text{Perimeter of Base} \times \text{Height of Prism} \]
\[ \text{Perimeter of Base} = 15 + 17 + 10 = 42 \, \text{cm} \]
\[ \text{Surface Area} = 2 \times 74.46 + 42 \times 8 \]
\[ \text{Surface Area} = 148.92 + 336 \]
\[ \text{Surface Area} = 484.92 \, \text{cm}^2 \]

Results:
- Volume = \( \approx 595.68 \, \text{cm}^3 \)
- Surface Area = \( \approx 484.92 \, \text{cm}^2 \)

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5. Square Pyramid

#### Dimensions:
- Base side (\( a \)) = 4 yd
- Slant height (\( l \)) = 10 yd

##### Step 1: Calculate the volume:
\[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

First, find the height (\( h \)) of the pyramid using the Pythagorean theorem in the right triangle formed by the height, half the base side, and the slant height:
\[ l^2 = \left(\frac{a}{2}\right)^2 + h^2 \]
\[ 10^2 = \left(\frac{4}{2}\right)^2 + h^2 \]
\[ 100 = 2^2 + h^2 \]
\[ 100 = 4 + h^2 \]
\[ h^2 = 96 \]
\[ h = \sqrt{96} = 4\sqrt{6} \, \text{yd} \]

Now, calculate the volume:
\[ \text{Base Area} = a^2 = 4^2 = 16 \, \text{yd}^2 \]
\[ \text{Volume} = \frac{1}{3} \times 16 \times 4\sqrt{6} \]
\[ \text{Volume} = \frac{64\sqrt{6}}{3} \, \text{yd}^3 \]

##### Step 2: Calculate the surface area:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]
\[ \text{Lateral Area} = 4 \times \left(\frac{1}{2} \times a \times l\right) \]
\[ \text{Lateral Area} = 4 \times \left(\frac{1}{2} \times 4 \times 10\right) \]
\[ \text{Lateral Area} = 4 \times 20 = 80 \, \text{yd}^2 \]
\[ \text{Surface Area} = 16 + 80 = 96 \, \text{yd}^2 \]

Results:
- Volume = \( \frac{64\sqrt{6}}{3} \, \text{yd}^3 \)
- Surface Area = \( 96 \, \text{yd}^2 \)

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6. Triangular Pyramid

#### Dimensions:
- Base triangle sides: 8 cm, 6 cm, 10 cm
- Height of pyramid (\( h \)) = 10 cm

##### Step 1: Calculate the volume:
\[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

First, find the area of the triangular base using Heron's formula:
\[ s = \frac{8 + 6 + 10}{2} = 12 \, \text{cm} \]
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
\[ \text{Area} = \sqrt{12(12-8)(12-6)(12-10)} \]
\[ \text{Area} = \sqrt{12 \times 4 \times 6 \times 2} \]
\[ \text{Area} = \sqrt{576} = 24 \, \text{cm}^2 \]

Now, calculate the volume:
\[ \text{Volume} = \frac{1}{3} \times 24 \times 10 \]
\[ \text{Volume} = 80 \, \text{cm}^3 \]

##### Step 2: Calculate the surface area:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]

The lateral area requires the slant heights of the triangular faces, which are not directly given. For simplicity, we assume the lateral area is not explicitly calculable without additional information. Thus, we only provide the base area here:
\[ \text{Surface Area} \approx 24 \, \text{cm}^2 \] (base area only)

Results:
- Volume = \( 80 \, \text{cm}^3 \)
- Surface Area = \( \text{Base Area} = 24 \, \text{cm}^2 \) (lateral area not calculable)

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7. Triangular Prism

#### Dimensions:
- Base triangle sides: 8 m, 6 m, 10 m
- Height of prism (\( h \)) = 5 m

##### Step 1: Calculate the volume:
\[ \text{Volume} = \text{Base Area} \times \text{Height of Prism} \]

First, find the area of the triangular base using Heron's formula:
\[ s = \frac{8 + 6 + 10}{2} = 12 \, \text{m} \]
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
\[ \text{Area} = \sqrt{12(12-8)(12-6)(12-10)} \]
\[ \text{Area} = \sqrt{12 \times 4 \times 6 \times 2} \]
\[ \text{Area} = \sqrt{576} = 24 \, \text{m}^2 \]

Now, calculate the volume:
\[ \text{Volume} = 24 \times 5 \]
\[ \text{Volume} = 120 \, \text{m}^3 \]

##### Step 2: Calculate the surface area:
\[ \text{Surface Area} = 2 \times \text{Base Area} + \text{Perimeter of Base} \times \text{Height of Prism} \]
\[ \text{Perimeter of Base} = 8 + 6 + 10 = 24 \, \text{m} \]
\[ \text{Surface Area} = 2 \times 24 + 24 \times 5 \]
\[ \text{Surface Area} = 48 + 120 \]
\[ \text{Surface Area} = 168 \, \text{m}^2 \]

Results:
- Volume = \( 120 \, \text{m}^3 \)
- Surface Area = \( 168 \, \text{m}^2 \)

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8. Cylinder

#### Dimensions:
- Radius (\( r \)) = 10 cm
- Height (\( h \)) = 8 cm

#### Volume:
\[ \text{Volume} = \pi r^2 h \]
\[ \text{Volume} = \pi \times 10^2 \times 8 \]
\[ \text{Volume} = \pi \times 100 \times 8 \]
\[ \text{Volume} = 800\pi \, \text{cm}^3 \]
\[ \text{Volume} \approx 2513.3 \, \text{cm}^3 \]

#### Surface Area:
\[ \text{Surface Area} = 2\pi r(h + r) \]
\[ \text{Surface Area} = 2\pi \times 10 \times (8 + 10) \]
\[ \text{Surface Area} = 2\pi \times 10 \times 18 \]
\[ \text{Surface Area} = 360\pi \, \text{cm}^2 \]
\[ \text{Surface Area} \approx 1131.0 \, \text{cm}^2 \]

Results:
- Volume = \( 800\pi \, \text{cm}^3 \) or \( \approx 2513.3 \, \text{cm}^3 \)
- Surface Area = \( 360\pi \, \text{cm}^2 \) or \( \approx 1131.0 \, \text{cm}^2 \)

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Final Answers:

1. Rectangular Prism:
- Volume = \( 180 \, \text{mm}^3 \)
- Surface Area = \( 216 \, \text{mm}^2 \)

2. Cube:
- Volume = \( 64 \, \text{mm}^3 \)
- Surface Area = \( 96 \, \text{mm}^2 \)

3. Cylinder:
- Volume = \( 490\pi \, \text{cm}^3 \) or \( \approx 1539.4 \, \text{cm}^3 \)
- Surface Area = \( 238\pi \, \text{cm}^2 \) or \( \approx 748.7 \, \text{cm}^2 \)

4. Triangular Prism:
- Volume = \( \approx 595.68 \, \text{cm}^3 \)
- Surface Area = \( \approx 484.92 \, \text{cm}^2 \)

5. Square Pyramid:
- Volume = \( \frac{64\sqrt{6}}{3} \, \text{yd}^3 \)
- Surface Area = \( 96 \, \text{yd}^2 \)

6. Triangular Pyramid:
- Volume = \( 80 \, \text{cm}^3 \)
- Surface Area = \( \text{Base Area} = 24 \, \text{cm}^2 \) (lateral area not calculable)

7. Triangular Prism:
- Volume = \( 120 \, \text{m}^3 \)
- Surface Area = \( 168 \, \text{m}^2 \)

8. Cylinder:
- Volume = \( 800\pi \, \text{cm}^3 \) or \( \approx 2513.3 \, \text{cm}^3 \)
- Surface Area = \( 360\pi \, \text{cm}^2 \) or \( \approx 1131.0 \, \text{cm}^2 \)

\boxed{\text{See detailed solutions above}}