3D Volume and Surface

Grade 7 · geometry · 100 practice problems · read aloud

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3D Volume and Surface Area

📐 What is it and Why is it Useful?

Volume is the amount of space inside a 3D shape (like how much water a cup can hold). Surface Area is the total area of all the faces on the outside of the shape (like the amount of wrapping paper needed to cover a box). We use these for real-world tasks like figuring out how much a storage container can hold or how much paint you need for a room!

🛠️ Step-by-Step Guide

  1. Identify the Shape: Is it a rectangular prism, cube, or triangular prism?
  2. Find the Right Formula: Each shape has its own formula.
  3. Plug in the Numbers: Carefully substitute the given measurements into the formula.
  4. Calculate: Solve the math step-by-step.
  5. Add Units: Always write your answer with cubic units (cm³, m³) for volume and square units (cm², m²) for surface area.

📚 Visual Examples

Example 1: Cube

A cube has side length = 5 cm.

  • Volume: V = s³ = 5 cm × 5 cm × 5 cm = 125 cm³
  • Surface Area: SA = 6s² = 6 × (5 cm × 5 cm) = 6 × 25 = 150 cm²

Example 2: Rectangular Prism

A box has length = 8 m, width = 3 m, height = 2 m.

  • Volume: V = l × w × h = 8 m × 3 m × 2 m = 48 m³
  • Surface Area: SA = 2lw + 2lh + 2wh = 2(8×3) + 2(8×2) + 2(3×2) = 2(24) + 2(16) + 2(6) = 48 + 32 + 12 = 92 m²

⚠️ Common Mistakes

  • Mixing up Formulas: Volume is 3D (cubed), Surface Area is 2D (squared). Remember: Volume fills it, Surface Area covers it.
  • Forgetting Units: Writing "48" instead of "48 m³" is incomplete. The units tell the story!
  • Surface Area Calculation: Students often forget one pair of faces. For a rectangular prism, you must find the area of all three different faces and double them.

💡 Tips & Tricks

  • Memory Aid: For a cube's surface area, remember "The 6 Sides Square-d" (6s²).
  • Draw a Net: Unfolding the 3D shape into a 2D net makes it easier to see all the faces for surface area.
  • Check Your Work: Does your answer make sense? A volume should be smaller than a surface area for the same small object? Not always! But a room with 48 m³ volume and 92 m² surface area seems reasonable.

🎯 Practice Suggestions

The best way to learn is by doing!

  • Start with simple cubes and rectangular prisms (boxes).
  • Find the volume and surface area of everyday objects: a cereal box, a dice, a tissue box.
  • Use graph paper to draw the nets of shapes and calculate their surface area.
  • Practice word problems that ask, "How much wrapping paper?" or "How many cubic meters of sand?"

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 ∛(125) + √(144) = ?

Hint: First, determine the number that, when multiplied by itself three times, gives the value inside the cube root symbol. Then, find the positive number that, when multiplied by itself, equals the value under the square root symbol. Finally, add these two results together.

Show the answer

Answer: 17

  1. Understand the problem. We need to evaluate the cube root of 125 and the square root of 144, then add them together.
  2. Evaluate the cube root of 125. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. We ask: what number multiplied by itself three times equals 125? 5 × 5 × 5 = 25 × 5 = 125. So, ∛(125) = 5.
  3. Evaluate the square root of 144. The square root of a number is a value that, when multiplied by itself, gives the original number. We ask: what number multiplied by itself equals 144? 12 × 12 = 144. So, √(144) = 12.
  4. Add the two results. 5 + 12 = 17.
  5. Final answer. The result of ∛(125) + √(144) is 17.

2 ∛(125) + 4² - √(81) = ?

Hint: Evaluate each term separately by finding roots and powers before combining them through addition and subtraction.

Show the answer

Answer: 12

  1. Evaluate cube root of 125 125 = 5 × 5 × 5 So cube root of 125 = 5
  2. Evaluate 4 squared 4 × 4 = 16
  3. Evaluate square root of 81 81 = 9 × 9 So square root of 81 = 9
  4. Substitute back into the expression 5 + 16 − 9
  5. Perform addition and subtraction from left to right 5 + 16 = 21 21 − 9 = 12 Final answer: 12

Let's solve step by step. We have: cube root of 125 + 4 squared − square root of 81

3 A rectangular prism has dimensions 12 cm × 8 cm × 15 cm. Calculate its surface area.

Hint: To find the surface area of a box, calculate the area of each of the six rectangular faces and add them together. Remember that opposite faces have equal areas.

Show the answer

Answer: 792 cm²

  1. Understand the formula for surface area of a rectangular prism. The surface area (SA) is the sum of the areas of all six faces. Formula: SA = 2 * (length * width + length * height + width * height)
  2. Identify the dimensions. Length (l) = 12 cm Width (w) = 8 cm Height (h) = 15 cm
  3. Calculate the area of the three different types of rectangular faces. Area of first face type (l × w) = 12 × 8 = 96 cm² Area of second face type (l × h) = 12 × 15 = 180 cm² Area of third face type (w × h) = 8 × 15 = 120 cm²
  4. Add these three areas together. Sum = 96 + 180 + 120 = 396 cm²
  5. Multiply by 2 because there are two of each face type in a rectangular prism. Total surface area = 2 × 396 = 792 cm²
  6. State the final answer. The surface area of the rectangular prism is 792 cm².

Let's find the surface area of the rectangular prism step by step.

4 A cylinder has a radius of 7 cm and a height of 12 cm. Find its volume and total surface area. Use π = 22/7.

Hint: Think about the formulas for a cylinder: volume uses the area of the circular base times the height, and surface area includes the two circular ends plus the curved side (which is a rectangle when unrolled).

Show the answer

Answer: Volume = 1848 cm³, Surface Area = 836 cm²

  1. Volume formula for a cylinder is V = πr²h. Substitute r = 7, h = 12, π = 22/7. V = (22/7) × 7² × 12 V = (22/7) × 49 × 12 V = 22 × 7 × 12 V = 22 × 84 V = 1848 cm³
  2. Surface area formula for a cylinder is SA = 2πr² + 2πrh. First, find 2πr² = 2 × (22/7) × 7² = 2 × (22/7) × 49 = 2 × 22 × 7 = 308 cm² Next, find 2πrh = 2 × (22/7) × 7 × 12 = 2 × 22 × 12 = 528 cm² Add them: SA = 308 + 528 = 836 cm² The volume is 1848 cm³ and the surface area is 836 cm².

5 A cylinder has a radius of 9 cm and a height of 15 cm. Find its volume and total surface area. Use π = 22/7.

Hint: Think about the formulas: volume is the area of the circular base times the height. Surface area includes the two circular ends and the curved side, which is a rectangle when unrolled. Use π = 22/7 for exact fractions.

Show the answer

Answer: Volume = 3811.5 cm³, Surface Area = 1357.71 cm²

  1. Volume formula for a cylinder is V = πr²h. Substitute r = 9, h = 15, π = 22/7. V = (22/7) × 9² × 15 V = (22/7) × 81 × 15 V = (22 × 81 × 15) / 7 V = (22 × 1215) / 7 V = 26730 / 7 V = 3818.57 cm³ (approximately)
  2. Surface area formula is SA = 2πr² + 2πrh. First, find 2πr² = 2 × (22/7) × 9² = 2 × (22/7) × 81 = (2 × 22 × 81) / 7 = 3564 / 7 = 509.14 cm² Next, find 2πrh = 2 × (22/7) × 9 × 15 = (2 × 22 × 9 × 15) / 7 = (44 × 135) / 7 = 5940 / 7 = 848.57 cm² Add them: SA = 509.14 + 848.57 = 1357.71 cm² The volume is 3818.57 cm³ and the surface area is 1357.71 cm².

6 A cylinder has a radius of 14 cm and a height of 25 cm. Find its volume and total surface area. Use π = 22/7.

Hint: Think about the two formulas you need. For volume, you multiply the area of the circular base by the height. For surface area, you need the area of the two circular ends plus the curved side — the curved side is like a rectangle when you unroll it, with width equal to the circumference of the circle.

Show the answer

Answer: Volume = 15400 cm³, Surface Area = 3432 cm²

  1. Volume formula for a cylinder is V = πr²h. Substitute r = 14, h = 25, π = 22/7. V = (22/7) × 14² × 25 V = (22/7) × 196 × 25 V = 22 × 28 × 25 V = 22 × 700 V = 15400 cm³
  2. Surface area formula for a cylinder is SA = 2πr² + 2πrh. First, find 2πr² = 2 × (22/7) × 14² = 2 × (22/7) × 196 = 2 × 22 × 28 = 1232 cm² Next, find 2πrh = 2 × (22/7) × 14 × 25 = 2 × 22 × 2 × 25 = 2 × 22 × 50 = 2200 cm² Add them: SA = 1232 + 2200 = 3432 cm² The volume is 15400 cm³ and the surface area is 3432 cm².
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