Math worksheet for calculating the volume of prisms, pyramids, cylinders, and cones.
Worksheet titled "Volume of Prisms, Pyramids, Cylinders, and Cones" with nine problems showing various 3D shapes including cylinders, rectangular prisms, pyramids, and cones, each with dimensions and space to calculate volume.
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Step-by-step solution for: Volume Formulas for Prisms, Pyramids, Cylinders, and Cones | PDF ...
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Show Answer Key & Explanations
Step-by-step solution for: Volume Formulas for Prisms, Pyramids, Cylinders, and Cones | PDF ...
To solve the problem, we need to calculate the volumes of various 3D shapes using their respective volume formulas. Let's go through each figure step by step.
The formula for the volume of a cylinder is:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
- Radius (\( r \)) = 7 yd
- Height (\( h \)) = 10 yd
\[
V = \pi (7)^2 (10) = \pi (49)(10) = 490\pi \approx 1539.38 \text{ yd}^3
\]
The formula for the volume of a rectangular prism is:
\[
V = l \times w \times h
\]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
- Length (\( l \)) = 14 in
- Width (\( w \)) = 6 in
- Height (\( h \)) = 8 in
\[
V = 14 \times 6 \times 8 = 672 \text{ in}^3
\]
The formula for the volume of a cube is:
\[
V = s^3
\]
where \( s \) is the side length.
- Side length (\( s \)) = 6 yd
\[
V = 6^3 = 216 \text{ yd}^3
\]
The formula for the volume of a pyramid is:
\[
V = \frac{1}{3} B h
\]
where \( B \) is the area of the base and \( h \) is the height.
- Base is a triangle with sides 11 ft and 4 ft.
- Height (\( h \)) = 13 ft
First, find the area of the triangular base using Heron's formula. The semi-perimeter \( s \) is:
\[
s = \frac{11 + 4 + 13}{2} = 14 \text{ ft}
\]
The area \( B \) is:
\[
B = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{14(14-11)(14-4)(14-13)} = \sqrt{14 \cdot 3 \cdot 10 \cdot 1} = \sqrt{420} \approx 20.49 \text{ ft}^2
\]
Now, calculate the volume:
\[
V = \frac{1}{3} \times 20.49 \times 13 \approx \frac{1}{3} \times 266.37 \approx 88.79 \text{ ft}^3
\]
The formula for the volume of a pyramid is:
\[
V = \frac{1}{3} B h
\]
where \( B \) is the area of the base and \( h \) is the height.
- Base is a triangle with sides 14 mm, 8 mm, and 6 mm.
- Height (\( h \)) = 6 mm
First, find the area of the triangular base using Heron's formula. The semi-perimeter \( s \) is:
\[
s = \frac{14 + 8 + 6}{2} = 14 \text{ mm}
\]
The area \( B \) is:
\[
B = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{14(14-14)(14-8)(14-6)} = \sqrt{14 \cdot 0 \cdot 6 \cdot 8} = 0 \text{ mm}^2
\]
This indicates an error in the problem setup or dimensions. Assuming correct dimensions, recheck the base area calculation.
The formula for the volume of a triangular prism is:
\[
V = B \times h
\]
where \( B \) is the area of the triangular base and \( h \) is the height of the prism.
- Base is an equilateral triangle with side 6 in.
- Height of the prism (\( h \)) = 12 in
First, find the area of the equilateral triangle base:
\[
B = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} (6)^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \approx 15.59 \text{ in}^2
\]
Now, calculate the volume:
\[
V = 15.59 \times 12 \approx 187.08 \text{ in}^3
\]
The formula for the volume of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
- Radius (\( r \)) = 3 cm
- Height (\( h \)) = 11 cm
\[
V = \frac{1}{3} \pi (3)^2 (11) = \frac{1}{3} \pi (9)(11) = 33\pi \approx 103.67 \text{ cm}^3
\]
The formula for the volume of a pyramid is:
\[
V = \frac{1}{3} B h
\]
where \( B \) is the area of the base and \( h \) is the height.
- Base is a square with side 14 ft.
- Height (\( h \)) = 10.954 ft
First, find the area of the square base:
\[
B = 14^2 = 196 \text{ ft}^2
\]
Now, calculate the volume:
\[
V = \frac{1}{3} \times 196 \times 10.954 \approx \frac{1}{3} \times 2147.624 \approx 715.87 \text{ ft}^3
\]
The formula for the volume of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
- Radius (\( r \)) = 4 mm
- Height (\( h \)) = 14 mm
\[
V = \frac{1}{3} \pi (4)^2 (14) = \frac{1}{3} \pi (16)(14) = \frac{224}{3}\pi \approx 234.56 \text{ mm}^3
\]
\[
\boxed{
\begin{array}{ll}
1) & 1539.38 \text{ yd}^3 \\
2) & 672 \text{ in}^3 \\
3) & 216 \text{ yd}^3 \\
4) & 88.79 \text{ ft}^3 \\
5) & \text{(Recheck dimensions)} \\
6) & 187.08 \text{ in}^3 \\
7) & 103.67 \text{ cm}^3 \\
8) & 715.87 \text{ ft}^3 \\
9) & 234.56 \text{ mm}^3 \\
\end{array}
}
\]
1) Cylinder
The formula for the volume of a cylinder is:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
- Radius (\( r \)) = 7 yd
- Height (\( h \)) = 10 yd
\[
V = \pi (7)^2 (10) = \pi (49)(10) = 490\pi \approx 1539.38 \text{ yd}^3
\]
2) Rectangular Prism
The formula for the volume of a rectangular prism is:
\[
V = l \times w \times h
\]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
- Length (\( l \)) = 14 in
- Width (\( w \)) = 6 in
- Height (\( h \)) = 8 in
\[
V = 14 \times 6 \times 8 = 672 \text{ in}^3
\]
3) Cube
The formula for the volume of a cube is:
\[
V = s^3
\]
where \( s \) is the side length.
- Side length (\( s \)) = 6 yd
\[
V = 6^3 = 216 \text{ yd}^3
\]
4) Pyramid
The formula for the volume of a pyramid is:
\[
V = \frac{1}{3} B h
\]
where \( B \) is the area of the base and \( h \) is the height.
- Base is a triangle with sides 11 ft and 4 ft.
- Height (\( h \)) = 13 ft
First, find the area of the triangular base using Heron's formula. The semi-perimeter \( s \) is:
\[
s = \frac{11 + 4 + 13}{2} = 14 \text{ ft}
\]
The area \( B \) is:
\[
B = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{14(14-11)(14-4)(14-13)} = \sqrt{14 \cdot 3 \cdot 10 \cdot 1} = \sqrt{420} \approx 20.49 \text{ ft}^2
\]
Now, calculate the volume:
\[
V = \frac{1}{3} \times 20.49 \times 13 \approx \frac{1}{3} \times 266.37 \approx 88.79 \text{ ft}^3
\]
5) Pyramid
The formula for the volume of a pyramid is:
\[
V = \frac{1}{3} B h
\]
where \( B \) is the area of the base and \( h \) is the height.
- Base is a triangle with sides 14 mm, 8 mm, and 6 mm.
- Height (\( h \)) = 6 mm
First, find the area of the triangular base using Heron's formula. The semi-perimeter \( s \) is:
\[
s = \frac{14 + 8 + 6}{2} = 14 \text{ mm}
\]
The area \( B \) is:
\[
B = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{14(14-14)(14-8)(14-6)} = \sqrt{14 \cdot 0 \cdot 6 \cdot 8} = 0 \text{ mm}^2
\]
This indicates an error in the problem setup or dimensions. Assuming correct dimensions, recheck the base area calculation.
6) Triangular Prism
The formula for the volume of a triangular prism is:
\[
V = B \times h
\]
where \( B \) is the area of the triangular base and \( h \) is the height of the prism.
- Base is an equilateral triangle with side 6 in.
- Height of the prism (\( h \)) = 12 in
First, find the area of the equilateral triangle base:
\[
B = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} (6)^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \approx 15.59 \text{ in}^2
\]
Now, calculate the volume:
\[
V = 15.59 \times 12 \approx 187.08 \text{ in}^3
\]
7) Cone
The formula for the volume of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
- Radius (\( r \)) = 3 cm
- Height (\( h \)) = 11 cm
\[
V = \frac{1}{3} \pi (3)^2 (11) = \frac{1}{3} \pi (9)(11) = 33\pi \approx 103.67 \text{ cm}^3
\]
8) Pyramid
The formula for the volume of a pyramid is:
\[
V = \frac{1}{3} B h
\]
where \( B \) is the area of the base and \( h \) is the height.
- Base is a square with side 14 ft.
- Height (\( h \)) = 10.954 ft
First, find the area of the square base:
\[
B = 14^2 = 196 \text{ ft}^2
\]
Now, calculate the volume:
\[
V = \frac{1}{3} \times 196 \times 10.954 \approx \frac{1}{3} \times 2147.624 \approx 715.87 \text{ ft}^3
\]
9) Cone
The formula for the volume of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height.
- Radius (\( r \)) = 4 mm
- Height (\( h \)) = 14 mm
\[
V = \frac{1}{3} \pi (4)^2 (14) = \frac{1}{3} \pi (16)(14) = \frac{224}{3}\pi \approx 234.56 \text{ mm}^3
\]
Final Answers:
\[
\boxed{
\begin{array}{ll}
1) & 1539.38 \text{ yd}^3 \\
2) & 672 \text{ in}^3 \\
3) & 216 \text{ yd}^3 \\
4) & 88.79 \text{ ft}^3 \\
5) & \text{(Recheck dimensions)} \\
6) & 187.08 \text{ in}^3 \\
7) & 103.67 \text{ cm}^3 \\
8) & 715.87 \text{ ft}^3 \\
9) & 234.56 \text{ mm}^3 \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of volumes of prisms and cylinders worksheet.
