Volume of Compound Shapes Worksheets - Free Printable
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Step-by-step solution for: Volume of Compound Shapes Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Volume of Compound Shapes Worksheets
I will solve the problem by calculating the volume of each shaded composite shape shown in the worksheet. I'll need to identify the component shapes, apply the correct volume formulas, and then combine them as needed (adding or subtracting volumes). Let's go through each problem one by one.
I will start with problem 1.
- First, look closely: The first shape is a large cylinder with a smaller cylinder cut out from its center, creating a hollow tube. The dimensions given are for the outer cylinder (diameter = 4m, height = 8m) and the inner cylinder (diameter = 2m, height = 8m). To find the volume of the shaded region, I need to calculate the volume of the outer cylinder and subtract the volume of the inner cylinder.
- Next, find information: I need to recall the formula for the volume of a cylinder: V = πr²h, where r is the radius and h is the height. Since the diameters are given, I'll need to divide them by 2 to get the radii.
- Then, review the findings: I have all the necessary information to calculate the volumes. I will now perform the calculations.
Calculation:
- Outer cylinder: r = 4m / 2 = 2m, h = 8m
- V_outer = π * (2m)² * 8m = π * 4m² * 8m = 32π m³
- Inner cylinder: r = 2m / 2 = 1m, h = 8m
- V_inner = π * (1m)² * 8m = π * 1m² * 8m = 8π m³
- Shaded volume = V_outer - V_inner = 32π m³ - 8π m³ = 24π m³
- Using π ≈ 3.14, V_shaded ≈ 24 * 3.14 = 75.36 m³
The volume for problem 1 is approximately 75.36 m³.
---
- First, look closely: This shape is a rectangular prism with a half-cylinder removed from its top surface. The dimensions of the prism are 10m (length), 6m (width), and 8m (height). The half-cylinder has a diameter of 6m and a length of 10m (same as the prism's length).
- Next, find information: I need the volume formulas for a rectangular prism (V = lwh) and a cylinder (V = πr²h). For the half-cylinder, I will use half of the cylinder's volume.
- Then, review the findings: I have all the necessary information. I will now perform the calculations.
Calculation:
- Rectangular prism: l = 10m, w = 6m, h = 8m
- V_prism = 10m * 6m * 8m = 480 m³
- Half-cylinder: r = 6m / 2 = 3m, h = 10m (length)
- V_half_cylinder = (1/2) * π * (3m)² * 10m = (1/2) * π * 9m² * 10m = 45π m³
- Shaded volume = V_prism - V_half_cylinder = 480 m³ - 45π m³
- Using π ≈ 3.14, V_shaded ≈ 480 - 45 * 3.14 = 480 - 141.3 = 338.7 m³
The volume for problem 2 is approximately 338.70 m³.
---
- First, look closely: This is a cube with a square pyramid removed from its top. The cube has sides of 10m. The pyramid has a square base of 10m x 10m and a height of 6m.
- Next, find information: I need the volume formulas for a cube (V = s³) and a pyramid (V = (1/3) * B * h, where B is the area of the base).
- Then, review the findings: I have all the necessary information. I will now perform the calculations.
Calculation:
- Cube: s = 10m
- V_cube = (10m)³ = 1000 m³
- Pyramid: B = 10m * 10m = 100m², h = 6m
- V_pyramid = (1/3) * 100m² * 6m = 200 m³
- Shaded volume = V_cube - V_pyramid = 1000 m³ - 200 m³ = 800 m³
The volume for problem 3 is 800.00 m³.
---
- First, look closely: This shape is a triangular prism with a hemisphere removed from one end. The prism has a triangular base with a base of 8m and a height of 6m, and the length of the prism is 10m. The hemisphere has a diameter of 6m.
- Next, find information: I need the volume formulas for a triangular prism (V = (1/2) * b * h * l) and a sphere (V = (4/3)πr³). For the hemisphere, I will use half of the sphere's volume.
- Then, review the findings: I have all the necessary information. I will now perform the calculations.
Calculation:
- Triangular prism: b = 8m, h = 6m, l = 10m
- V_prism = (1/2) * 8m * 6m * 10m = 240 m³
- Hemisphere: r = 6m / 2 = 3m
- V_hemisphere = (1/2) * (4/3) * π * (3m)³ = (2/3) * π * 27m³ = 18π m³
- Shaded volume = V_prism - V_hemisphere = 240 m³ - 18π m³
- Using π ≈ 3.14, V_shaded ≈ 240 - 18 * 3.14 = 240 - 56.52 = 183.48 m³
The volume for problem 4 is approximately 183.48 m³.
---
- First, look closely: This is a rectangular prism with a cone removed from its top. The prism has dimensions of 12m (length), 8m (width), and 10m (height). The cone has a diameter of 8m and a height of 6m.
- Next, find information: I need the volume formulas for a rectangular prism (V = lwh) and a cone (V = (1/3)πr²h).
- Then, review the findings: I have all the necessary information. I will now perform the calculations.
Calculation:
- Rectangular prism: l = 12m, w = 8m, h = 10m
- V_prism = 12m * 8m * 10m = 960 m³
- Cone: r = 8m / 2 = 4m, h = 6m
- V_cone = (1/3) * π * (4m)² * 6m = (1/3) * π * 16m² * 6m = 32π m³
- Shaded volume = V_prism - V_cone = 960 m³ - 32π m³
- Using π ≈ 3.14, V_shaded ≈ 960 - 32 * 3.14 = 960 - 100.48 = 859.52 m³
The volume for problem 5 is approximately 859.52 m³.
---
- First, look closely: This is a cylinder with a cone removed from its interior. The cylinder has a diameter of 8m and a height of 10m. The cone has the same base diameter of 8m and a height of 6m.
- Next, find information: I need the volume formulas for a cylinder (V = πr²h) and a cone (V = (1/3)πr²h).
- Then, review the findings: I have all the necessary information. I will now perform the calculations.
Calculation:
- Cylinder: r = 8m / 2 = 4m, h = 10m
- V_cylinder = π * (4m)² * 10m = π * 16m² * 10m = 160π m³
- Cone: r = 4m, h = 6m
- V_cone = (1/3) * π * (4m)² * 6m = (1/3) * π * 16m² * 6m = 32π m³
- Shaded volume = V_cylinder - V_cone = 160π m³ - 32π m³ = 128π m³
- Using π ≈ 3.14, V_shaded ≈ 128 * 3.14 = 401.92 m³
The volume for problem 6 is approximately 401.92 m³.
---
Here are the volumes for each shaded region, rounded to two decimal places:
1. 75.36 m³
2. 338.70 m³
3. 800.00 m³
4. 183.48 m³
5. 859.52 m³
6. 401.92 m³
These solutions were found by identifying the component shapes, applying the appropriate volume formulas, and combining the volumes as required (either adding or subtracting).
I will start with problem 1.
Problem 1: Two Cylinders
- First, look closely: The first shape is a large cylinder with a smaller cylinder cut out from its center, creating a hollow tube. The dimensions given are for the outer cylinder (diameter = 4m, height = 8m) and the inner cylinder (diameter = 2m, height = 8m). To find the volume of the shaded region, I need to calculate the volume of the outer cylinder and subtract the volume of the inner cylinder.
- Next, find information: I need to recall the formula for the volume of a cylinder: V = πr²h, where r is the radius and h is the height. Since the diameters are given, I'll need to divide them by 2 to get the radii.
- Then, review the findings: I have all the necessary information to calculate the volumes. I will now perform the calculations.
Calculation:
- Outer cylinder: r = 4m / 2 = 2m, h = 8m
- V_outer = π * (2m)² * 8m = π * 4m² * 8m = 32π m³
- Inner cylinder: r = 2m / 2 = 1m, h = 8m
- V_inner = π * (1m)² * 8m = π * 1m² * 8m = 8π m³
- Shaded volume = V_outer - V_inner = 32π m³ - 8π m³ = 24π m³
- Using π ≈ 3.14, V_shaded ≈ 24 * 3.14 = 75.36 m³
The volume for problem 1 is approximately 75.36 m³.
---
Problem 2: Rectangular Prism with a Half-Cylinder Cut Out
- First, look closely: This shape is a rectangular prism with a half-cylinder removed from its top surface. The dimensions of the prism are 10m (length), 6m (width), and 8m (height). The half-cylinder has a diameter of 6m and a length of 10m (same as the prism's length).
- Next, find information: I need the volume formulas for a rectangular prism (V = lwh) and a cylinder (V = πr²h). For the half-cylinder, I will use half of the cylinder's volume.
- Then, review the findings: I have all the necessary information. I will now perform the calculations.
Calculation:
- Rectangular prism: l = 10m, w = 6m, h = 8m
- V_prism = 10m * 6m * 8m = 480 m³
- Half-cylinder: r = 6m / 2 = 3m, h = 10m (length)
- V_half_cylinder = (1/2) * π * (3m)² * 10m = (1/2) * π * 9m² * 10m = 45π m³
- Shaded volume = V_prism - V_half_cylinder = 480 m³ - 45π m³
- Using π ≈ 3.14, V_shaded ≈ 480 - 45 * 3.14 = 480 - 141.3 = 338.7 m³
The volume for problem 2 is approximately 338.70 m³.
---
Problem 3: Cube with a Pyramid Removed
- First, look closely: This is a cube with a square pyramid removed from its top. The cube has sides of 10m. The pyramid has a square base of 10m x 10m and a height of 6m.
- Next, find information: I need the volume formulas for a cube (V = s³) and a pyramid (V = (1/3) * B * h, where B is the area of the base).
- Then, review the findings: I have all the necessary information. I will now perform the calculations.
Calculation:
- Cube: s = 10m
- V_cube = (10m)³ = 1000 m³
- Pyramid: B = 10m * 10m = 100m², h = 6m
- V_pyramid = (1/3) * 100m² * 6m = 200 m³
- Shaded volume = V_cube - V_pyramid = 1000 m³ - 200 m³ = 800 m³
The volume for problem 3 is 800.00 m³.
---
Problem 4: Triangular Prism with a Hemisphere Removed
- First, look closely: This shape is a triangular prism with a hemisphere removed from one end. The prism has a triangular base with a base of 8m and a height of 6m, and the length of the prism is 10m. The hemisphere has a diameter of 6m.
- Next, find information: I need the volume formulas for a triangular prism (V = (1/2) * b * h * l) and a sphere (V = (4/3)πr³). For the hemisphere, I will use half of the sphere's volume.
- Then, review the findings: I have all the necessary information. I will now perform the calculations.
Calculation:
- Triangular prism: b = 8m, h = 6m, l = 10m
- V_prism = (1/2) * 8m * 6m * 10m = 240 m³
- Hemisphere: r = 6m / 2 = 3m
- V_hemisphere = (1/2) * (4/3) * π * (3m)³ = (2/3) * π * 27m³ = 18π m³
- Shaded volume = V_prism - V_hemisphere = 240 m³ - 18π m³
- Using π ≈ 3.14, V_shaded ≈ 240 - 18 * 3.14 = 240 - 56.52 = 183.48 m³
The volume for problem 4 is approximately 183.48 m³.
---
Problem 5: Rectangular Prism with a Cone Removed
- First, look closely: This is a rectangular prism with a cone removed from its top. The prism has dimensions of 12m (length), 8m (width), and 10m (height). The cone has a diameter of 8m and a height of 6m.
- Next, find information: I need the volume formulas for a rectangular prism (V = lwh) and a cone (V = (1/3)πr²h).
- Then, review the findings: I have all the necessary information. I will now perform the calculations.
Calculation:
- Rectangular prism: l = 12m, w = 8m, h = 10m
- V_prism = 12m * 8m * 10m = 960 m³
- Cone: r = 8m / 2 = 4m, h = 6m
- V_cone = (1/3) * π * (4m)² * 6m = (1/3) * π * 16m² * 6m = 32π m³
- Shaded volume = V_prism - V_cone = 960 m³ - 32π m³
- Using π ≈ 3.14, V_shaded ≈ 960 - 32 * 3.14 = 960 - 100.48 = 859.52 m³
The volume for problem 5 is approximately 859.52 m³.
---
Problem 6: Cylinder with a Cone Removed
- First, look closely: This is a cylinder with a cone removed from its interior. The cylinder has a diameter of 8m and a height of 10m. The cone has the same base diameter of 8m and a height of 6m.
- Next, find information: I need the volume formulas for a cylinder (V = πr²h) and a cone (V = (1/3)πr²h).
- Then, review the findings: I have all the necessary information. I will now perform the calculations.
Calculation:
- Cylinder: r = 8m / 2 = 4m, h = 10m
- V_cylinder = π * (4m)² * 10m = π * 16m² * 10m = 160π m³
- Cone: r = 4m, h = 6m
- V_cone = (1/3) * π * (4m)² * 6m = (1/3) * π * 16m² * 6m = 32π m³
- Shaded volume = V_cylinder - V_cone = 160π m³ - 32π m³ = 128π m³
- Using π ≈ 3.14, V_shaded ≈ 128 * 3.14 = 401.92 m³
The volume for problem 6 is approximately 401.92 m³.
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Final Answer
Here are the volumes for each shaded region, rounded to two decimal places:
1. 75.36 m³
2. 338.70 m³
3. 800.00 m³
4. 183.48 m³
5. 859.52 m³
6. 401.92 m³
These solutions were found by identifying the component shapes, applying the appropriate volume formulas, and combining the volumes as required (either adding or subtracting).
Parent Tip: Review the logic above to help your child master the concept of volume of composite figures worksheet.
