Volume Applications

Grade 8 · geometry · 102 practice problems · read aloud

🔊 Listen to this explanation

📦 What is Volume and Why is it Useful?

Volume is the amount of 3-dimensional space an object occupies. We measure it in cubic units (like cm³, m³). It's useful for real-world tasks like figuring out how much water fills a pool, how much a box can hold, or the amount of concrete needed for a foundation.

🧩 Step-by-Step Guide to Finding Volume

  1. Identify the Shape: Is it a rectangular prism, cylinder, or triangular prism?
  2. Find the Formula: Each 3D shape has a specific volume formula.
  3. Measure the Dimensions: Carefully find the needed lengths (length, width, height, radius).
  4. Plug and Calculate: Substitute the numbers into the formula and solve.
  5. Add Units: Always write your answer with cubic units (e.g., cm³).

🔢 Visual Examples

Example 1: Rectangular Prism (Box)

A box has a length of 5 cm, width of 3 cm, and height of 4 cm.

Formula: V = l × w × h

Calculation: V = 5 cm × 3 cm × 4 cm = 60 cm³

Example 2: Cylinder (Can)

A can has a radius of 2 m and a height of 7 m.

Formula: V = πr²h

Calculation: V = 3.14 × (2 m)² × 7 m = 3.14 × 4 m² × 7 m = 87.92 m³

⚠️ Common Mistakes to Avoid

  • Using Area Formulas: Don't confuse volume (3D) with surface area (the outside). Volume is inside space.
  • Mixing Units: Ensure all measurements are in the same unit before calculating (don't mix cm and m!).
  • Forgetting to Cube Units: Your final answer must be in cubic units. A volume of 60 is wrong; it's 60 cm³.
  • Cylinder Radius: Using the diameter instead of the radius in V = πr²h. Remember: radius = diameter ÷ 2.

💡 Tips & Tricks

  • Rectangular Prism: Think "length × width × height" as filling the base layer and then stacking layers upwards.
  • Cylinder: The formula is just (Area of the circular base) × height. V = (πr²) × h.
  • Estimate: Before you calculate, make a rough guess. Does your final answer seem reasonable for the object's size?

🎯 How to Practice

  • Find Volumes at Home: Calculate the volume of a cereal box, a water bottle, or your classroom.
  • Word Problems: Practice with problems that describe a real-world scenario.
  • Mix It Up: Create a set of flashcards with different shapes and formulas to test your memory.
  • Draw It: Sketch the 3D object and label its dimensions before you start calculating.

Practice problems

6 of the 102, worked through step by step — try them before opening the answer.

1 A rectangular prism has dimensions 2.5 × 4 × 6. What is its volume?

Hint: To find the volume of a rectangular prism, multiply its length, width, and height together.

Show the answer

Answer: 60

  1. Identify the dimensions. Length = 2.5 Width = 4 Height = 6
  2. Write the volume formula. Volume = Length × Width × Height
  3. Substitute the numbers into the formula. Volume = 2.5 × 4 × 6
  4. Perform the multiplication step by step. First, multiply 2.5 and 4. 2.5 × 4 = 10
  5. Now multiply the result (10) by the height (6). 10 × 6 = 60
  6. State the final answer. The volume of the rectangular prism is 60.

To find the volume of a rectangular prism, we multiply its three dimensions: length, width, and height.

2 A rectangular prism has length 3.5 m, width 2.8 m, and height 1.2 m. What is its volume?

Hint: Remember that volume of a rectangular prism is found by multiplying its three dimensions together. Pay attention to decimal placement.

Show the answer

Answer: 11.76

  1. Volume of a rectangular prism = length × width × height
  2. Substitute the given values: 3.5 × 2.8 × 1.2
  3. First multiply 3.5 × 2.8 = 9.8
  4. Then multiply 9.8 × 1.2 = 11.76
  5. The volume is 11.76 cubic meters

The answer is 11.76.

3 A rectangular prism has length 3.5 m, width 2.8 m, and height 4.2 m. What is its volume?

Hint: Volume of a rectangular prism is found by multiplying its three dimensions together. Make sure all measurements are in the same units before multiplying.

Show the answer

Answer: 41.16

  1. Write the volume formula for a rectangular prism: Volume = length × width × height
  2. Substitute the given values: Volume = 3.5 × 2.8 × 4.2
  3. First multiply 3.5 × 2.8 = 9.8
  4. Then multiply 9.8 × 4.2 = 41.16
  5. Include the units: 41.16 cubic meters The volume is 41.16 m³.

4 A spherical water tank has a diameter of 9 meters. What is its volume in cubic meters? (Use π ≈ 3.14)

Hint: Think about which formula to use for a sphere. Remember that the radius is half the diameter, and you need to cube the radius.

Show the answer

Answer: 381.51

  1. Find the radius from the diameter: radius = diameter ÷ 2 = 9 ÷ 2 = 4.5 meters
  2. Use the sphere volume formula: V = (4/3) × π × r³
  3. Substitute the values: V = (4/3) × 3.14 × (4.5)³
  4. Calculate (4.5)³ = 4.5 × 4.5 × 4.5 = 20.25 × 4.5 = 91.125
  5. Multiply by π: 3.14 × 91.125 = 286.1325
  6. Multiply by (4/3): (4/3) × 286.1325 = (4 × 286.1325) ÷ 3 = 1144.53 ÷ 3 = 381.51 The volume is 381.51 cubic meters.

5 A spherical water tank has a diameter of 6 meters. What is its volume in cubic meters? (Use π ≈ 3.14)

Hint: Recall the formula for the volume of a sphere. You need the radius, which is half the diameter. Cube the radius before multiplying by pi and the constant.

Show the answer

Answer: 113.04

  1. Find the radius: radius = diameter ÷ 2 = 6 ÷ 2 = 3 meters.
  2. Use the sphere volume formula: V = (4/3) × π × r³.
  3. Substitute the values: V = (4/3) × 3.14 × (3)³.
  4. Calculate the cube: 3³ = 27.
  5. Multiply: (4/3) × 3.14 × 27 = (4/3) × 84.78.
  6. Multiply: (4 × 84.78) ÷ 3 = 339.12 ÷ 3 = 113.04. The volume is 113.04 cubic meters.

6 A spherical water tank has a radius of 1.5 meters. What is its volume in cubic meters? (Use π ≈ 3.14)

Hint: Think about which formula applies to a sphere. You will need to cube the radius and multiply by pi and a special fraction.

Show the answer

Answer: 14.13

  1. The formula for the volume of a sphere is V = (4/3) × π × r³.
  2. Substitute the given values: r = 1.5 m, π ≈ 3.14.
  3. Calculate r³: 1.5³ = 1.5 × 1.5 × 1.5 = 2.25 × 1.5 = 3.375.
  4. Multiply by π: 3.14 × 3.375 = 10.5975.
  5. Multiply by 4/3: (4/3) × 10.5975 = (4 × 10.5975) / 3 = 42.39 / 3 = 14.13. The volume is 14.13 cubic meters.
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