Volume vs Surface Area

Show your work for each problem.

1.
You want to know how much water a fish tank can hold. Which measurement do you need?
1. a)Volume
2. b)Perimeter
3. c)Length
4. d)Surface area
Answer: Volume
Volume measures how much space is inside an object - exactly what you need to know for water capacity.
2.
You want to wrap a gift box. Which measurement tells you how much wrapping paper you need?
1. a)Diagonal
2. b)Volume
3. c)Weight
4. d)Surface area
Answer: Surface area
Surface area measures the total area of all outside surfaces - perfect for knowing how much paper covers a box.
3.
A cube has a side length of 5 cm. What is its volume in cm³?
Answer: 125
cm³
4.
A cube has a side length of 5 cm. What is its surface area in cm²?
Answer: 150
cm²
5.
What unit is used for volume?
1. a)cm (linear)
2. b)kg (mass)
3. c)cm² (square)
4. d)cm³ (cubic)
Answer: cm³ (cubic)
Volume measures 3D space, so we use cubic units (cm³, m³). Surface area uses square units (cm²).
6.
A rectangular box is 8 cm long, 5 cm wide, and 3 cm tall. Calculate both its volume and surface area.
Answer: V=120, SA=158
1. What is the formula for volume of a rectangular prism? l*w*h
2. Calculate the volume: 8 × 5 × 3 = ? 120
3. For surface area, we need 2(lw + lh + wh). What is lw (8 × 5)? 40
4. What is lh (8 × 3)? 24
5. What is wh (5 × 3)? 15
6. Calculate SA = 2(40 + 24 + 15) = ? 158
7.
A painter needs to paint the walls of a room. Which measurement helps determine how much paint to buy?
1. a)Height of ceiling
2. b)Volume of room
3. c)Surface area of walls
4. d)Number of windows
Answer: Surface area of walls
Paint covers surfaces, so we need surface area. Volume would tell us how much air is in the room, not how much paint we need.
8.
A box has dimensions 10 cm × 6 cm × 4 cm. What is its volume in cm³?
Answer: 240
cm³
9.
A box has dimensions 10 cm × 6 cm × 4 cm. What is its surface area in cm²?
Answer: 248
cm²
10.
Box A is 3×3×8 cm. Box B is 4×4×4.5 cm. Which has more volume? Which has more surface area?
Answer: B more volume, A more SA
1. Calculate volume of A: 3 × 3 × 8 = ? 72
2. Calculate volume of B: 4 × 4 × 4.5 = ? 72
3. Which has more volume? same
4. Calculate SA of A: 2(9 + 24 + 24) = ? 114
5. Calculate SA of B: 2(16 + 18 + 18) = ? 104
6. Which has more surface area? A
11.
An air conditioning company sizes units based on room measurements. Should they use volume or surface area?
1. a)Volume - to cool the air inside
2. b)Perimeter - to measure the edges
3. c)Surface area - to cover the walls
4. d)Height only - that's what matters
Answer: Volume - to cool the air inside
Air conditioning cools the air inside the room. Volume tells us how much air needs to be cooled.
12.
A swimming pool is 25m long, 10m wide, and 2m deep. How many liters of water does it hold? (1 m³ = 1000 liters)
Answer: 500000
1. What measurement do we need for water capacity? volume
2. Calculate length × width: 25 × 10 = ? 250
3. Multiply by depth: 250 × 2 = ? 500
4. Convert to liters: 500 × 1000 = ? 500000
13.
A warehouse floor is 50m × 30m. The walls are 8m high. What is the surface area of all 4 walls (not floor or ceiling) in m²?
Answer: 1280
Walls: m²
14.
A cereal box is 20 cm × 6 cm × 30 cm. The company wants to reduce cardboard cost by 10%. What would the new surface area be in cm²?
Answer: 1620
1. Calculate lw (20 × 6) = ? 120
2. Calculate lh (20 × 30) = ? 600
3. Calculate wh (6 × 30) = ? 180
4. Current SA = 2(120 + 600 + 180) = ? 1800
5. 10% of 1800 = ? 180
6. New SA = 1800 - 180 = ? 1620
15.
Which shape has the smallest surface area for a given volume?
1. a)Flat wide box
2. b)Long thin box
3. c)Cube
4. d)Sphere
Answer: Sphere
A sphere has the smallest surface area for any given volume. This is why bubbles, planets, and water droplets are spherical - nature minimizes surface tension!
16.
A juice box holds 250 mL. You want to know how much plastic covers the outside. Do you need volume or surface area?
Answer: surface area
1. What does '250 mL' tell us about? volume
2. What does 'plastic covering the outside' relate to? surface area
3. So which measurement do we need? surface area
17.
A gift box is 15 cm × 15 cm × 15 cm. How much wrapping paper (in cm²) is needed to cover all sides?
Answer: 1350
For a cube: cm²
18.
A storage container is 2m × 1.5m × 1m. You need to: (a) Find how many cubic meters it can hold. (b) Find how much material covers all 6 sides.
Answer: V=3, SA=13
1. (a) asks about capacity. Is that volume or surface area? volume
2. Calculate volume: 2 × 1.5 × 1 = ? 3
3. (b) asks about material covering. Is that volume or surface area? surface area
4. Calculate lw (2 × 1.5) = ? 3
5. Calculate lh (2 × 1) = ? 2
6. Calculate wh (1.5 × 1) = ? 1.5
7. SA = 2(3 + 2 + 1.5) = ? 13