Surface Area and Volume

Three-dimensional shapes are all around us! Understanding surface area and volume helps you solve real-world problems involving boxes, containers, buildings, and more. Let's explore how to calculate these important measurements!

---

3D Shapes vs. 2D Shapes

2D shapes are flat (like squares, circles, triangles)

• Measure: Area (square units)

3D shapes have length, width, AND height

• Measure: Volume (cubic units) and Surface Area (square units)

Common 3D shapes:

• Rectangular prism (box)
• Cube
• Cylinder
• Cone
• Sphere
• Pyramid

---

What Is Volume?

Volume measures the amount of space inside a 3D shape.

Think of it as: How much liquid could it hold? How many unit cubes fit inside?

Units: Cubic units (cm³, in³, m³, ft³)

Example: A box that is 3 cm × 4 cm × 2 cm has volume = 24 cm³ (24 cubic centimeters of space inside)

---

What Is Surface Area?

Surface Area measures the total area of all the outer surfaces of a 3D shape.

Think of it as: How much wrapping paper do you need to cover it? How much paint to cover all sides?

Units: Square units (cm², in², m², ft²)

Example: A cube with 6 faces, each 2 cm × 2 cm, has surface area = 6 × 4 = 24 cm²

---

Rectangular Prism (Box)

A box with length (l), width (w), and height (h).

Volume Formula

V = l × w × h

Example: Find the volume of a box: length = 5 ft, width = 3 ft, height = 4 ft

Solution: V = 5 × 3 × 4 = 60 ft³

Answer: 60 cubic feet

Surface Area Formula

A rectangular prism has 6 faces (top, bottom, front, back, left, right).

SA = 2lw + 2lh + 2wh

Or think: SA = 2(lw + lh + wh)

Example: Same box (5 ft × 3 ft × 4 ft)

Solution: SA = 2(5×3) + 2(5×4) + 2(3×4) SA = 2(15) + 2(20) + 2(12) SA = 30 + 40 + 24 SA = 94 ft²

Answer: 94 square feet

---

Cube

A cube is a special rectangular prism where all edges are equal (s = side length).

Volume Formula

V = s³ (side × side × side)

Example: A cube with side = 6 cm

Solution: V = 6³ = 6 × 6 × 6 = 216 cm³

Answer: 216 cubic centimeters

Surface Area Formula

A cube has 6 identical square faces.

SA = 6s²

Example: Same cube (side = 6 cm)

Solution: SA = 6 × (6²) = 6 × 36 = 216 cm²

Answer: 216 square centimeters

Note: For this cube, volume and surface area have the same number, but DIFFERENT UNITS!

---

Cylinder

A cylinder has two circular bases and a curved side.

Volume Formula

V = πr²h

Where r = radius of the base, h = height

Example: A cylinder with radius = 4 in and height = 10 in

Solution: V = π × (4²) × 10 V = π × 16 × 10 V = 160π ≈ 502.65 in³

Answer: About 503 cubic inches

Surface Area Formula

Surface area = 2 circular bases + curved side (which "unrolls" into a rectangle)

SA = 2πr² + 2πrh

Or: SA = 2πr(r + h)

Example: Same cylinder (r = 4 in, h = 10 in)

Solution: SA = 2π(4²) + 2π(4)(10) SA = 2π(16) + 2π(40) SA = 32π + 80π SA = 112π ≈ 351.86 in²

Answer: About 352 square inches

---

Triangular Prism

A prism with triangular bases.

Volume Formula

V = (Area of base) × height

For triangular base: V = (1/2 × base × height of triangle) × prism height

Example: Triangular prism with:

• Triangle base = 6 cm, triangle height = 4 cm
• Prism height = 10 cm

Solution: V = (1/2 × 6 × 4) × 10 V = (12) × 10 V = 120 cm³

Answer: 120 cubic centimeters

Surface Area

Add: 2 triangular bases + 3 rectangular sides

Example: Calculate each face area and add them all up!

---

Cone

A cone has one circular base and comes to a point (apex).

Volume Formula

V = (1/3)πr²h

Where r = radius, h = height (perpendicular from base to apex)

Example: A cone with radius = 3 ft and height = 8 ft

Solution: V = (1/3)π(3²)(8) V = (1/3)π(9)(8) V = (1/3)π(72) V = 24π ≈ 75.40 ft³

Answer: About 75 cubic feet

Note: Cone volume is 1/3 of a cylinder with same base and height!

---

Sphere

A perfectly round ball shape.

Volume Formula

V = (4/3)πr³

Where r = radius

Example: A sphere with radius = 5 m

Solution: V = (4/3)π(5³) V = (4/3)π(125) V = (500/3)π ≈ 523.60 m³

Answer: About 524 cubic meters

Surface Area Formula

SA = 4πr²

Example: Same sphere (r = 5 m)

Solution: SA = 4π(5²) SA = 4π(25) SA = 100π ≈ 314.16 m²

Answer: About 314 square meters

---

Pyramid

A pyramid has a base (often square) and triangular faces meeting at an apex.

Volume Formula

V = (1/3) × (Area of base) × height

For square base: V = (1/3) × s² × h

Example: Square pyramid with base side = 6 cm and height = 9 cm

Solution: V = (1/3) × (6²) × 9 V = (1/3) × 36 × 9 V = (1/3) × 324 V = 108 cm³

Answer: 108 cubic centimeters

Note: Pyramid volume is 1/3 of a prism with same base and height!

---

Real-World Applications

Packaging (Rectangular Prism)

Problem: A shipping box is 12 in × 8 in × 6 in. How much can it hold? How much cardboard is needed to make it?

Volume (capacity): V = 12 × 8 × 6 = 576 in³

Surface Area (cardboard): SA = 2(12×8) + 2(12×6) + 2(8×6) SA = 2(96) + 2(72) + 2(48) SA = 192 + 144 + 96 = 432 in²

Answer: Holds 576 in³, needs 432 in² of cardboard

Swimming Pool (Rectangular Prism)

Problem: A pool is 20 ft long, 10 ft wide, and 5 ft deep. How many gallons of water does it hold? (1 ft³ = 7.48 gallons)

Solution: V = 20 × 10 × 5 = 1,000 ft³ Gallons = 1,000 × 7.48 = 7,480 gallons

Answer: 7,480 gallons

Soda Can (Cylinder)

Problem: A can has radius 3 cm and height 12 cm. How much soda does it hold?

Solution: V = πr²h = π(3²)(12) = 108π ≈ 339.29 cm³

Answer: About 339 cm³ (about 339 mL)

Ice Cream Cone

Problem: An ice cream cone has radius 2 in and height 6 in. What's its volume?

Solution: V = (1/3)πr²h = (1/3)π(2²)(6) = 8π ≈ 25.13 in³

Answer: About 25 cubic inches

Basketball (Sphere)

Problem: A basketball has radius 4.7 inches. What's its volume?

Solution: V = (4/3)πr³ = (4/3)π(4.7³) ≈ 434.89 in³

Answer: About 435 cubic inches

---

Comparing Volume Formulas

Notice the patterns:

Prisms: V = (Base Area) × Height

• Rectangular: V = lwh
• Triangular: V = (1/2)bh × H
• Cylinder: V = πr²h

Pyramids/Cones: V = (1/3) × (Base Area) × Height

• Square pyramid: V = (1/3)s²h
• Cone: V = (1/3)πr²h

Pattern: Pyramids and cones are 1/3 the volume of prisms/cylinders with the same base and height!

Sphere: V = (4/3)πr³ (unique formula)

---

Finding Unknown Dimensions

Sometimes you know the volume and need to find a dimension!

Example 1: Find Height

A rectangular prism has volume 120 cm³, length = 6 cm, width = 4 cm. Find height.

Solution: V = lwh 120 = 6 × 4 × h 120 = 24h h = 5 cm

Answer: Height = 5 cm

Example 2: Find Radius

A cylinder has volume 150π in³ and height = 6 in. Find radius.

Solution: V = πr²h 150π = πr²(6) 150π = 6πr² 25 = r² r = 5 in

Answer: Radius = 5 inches

---

Units and Conversions

Important: Keep units consistent!

Volume units:

• 1 m³ = 1,000,000 cm³
• 1 ft³ = 1,728 in³
• 1 m³ = 1,000 liters
• 1 cm³ = 1 milliliter (mL)

Example: A box is 10 cm × 10 cm × 10 cm. Find volume in cm³ and liters.

Solution: V = 10³ = 1,000 cm³ In liters: 1,000 cm³ = 1,000 mL = 1 liter

Answer: 1,000 cm³ or 1 liter

---

Common Mistakes to Avoid

❌ Mistake 1: Confusing volume and surface area

• Volume = inside space (cubic units)
• Surface Area = outside covering (square units)

❌ Mistake 2: Forgetting the 1/3 for cones and pyramids

• Wrong: Cone V = πr²h
• Right: Cone V = (1/3)πr²h

❌ Mistake 3: Using diameter instead of radius

• Formulas use RADIUS (half of diameter)
• If given diameter 10, use r = 5!

❌ Mistake 4: Wrong units

• Volume needs CUBIC units (cm³, not cm²)
• Surface Area needs SQUARE units (cm², not cm³)

❌ Mistake 5: Calculation errors with π

• Use calculator's π button for accuracy
• Or use 3.14 as approximation

❌ Mistake 6: Mixing up dimensions

• Height vs. slant height (pyramids, cones)
• Base vs. total height
• Label your diagram!

---

Practice Tips

Tip 1: Draw and label

• Sketch the 3D shape
• Label all dimensions clearly
• Mark what you're finding

Tip 2: Identify the shape

• What 3D shape is it?
• Which formula do you need?

Tip 3: Check what's given

• Do you have radius or diameter?
• Are all measurements in same units?
• Convert if needed!

Tip 4: Use formulas correctly

• Write the formula first
• Substitute values
• Calculate step-by-step

Tip 5: Check reasonableness

• Volume should be in cubic units
• Surface area should be in square units
• Does the size make sense?

Tip 6: Remember special cases

• Cube: all edges equal
• Cylinder: circular bases
• Cone/pyramid: includes 1/3

---

Quick Reference Formulas

Rectangular Prism:

• Volume: V = lwh
• Surface Area: SA = 2(lw + lh + wh)

Cube:

• Volume: V = s³
• Surface Area: SA = 6s²

Cylinder:

• Volume: V = πr²h
• Surface Area: SA = 2πr² + 2πrh

Cone:

• Volume: V = (1/3)πr²h

Sphere:

• Volume: V = (4/3)πr³
• Surface Area: SA = 4πr²

Square Pyramid:

• Volume: V = (1/3)s²h

Triangular Prism:

• Volume: V = (1/2 × base × triangle height) × prism height

---

Problem-Solving Strategy

Step 1: Identify the 3D shape

Step 2: Determine what you're finding (volume or surface area)

Step 3: List what you know (dimensions)

Step 4: Write the appropriate formula

Step 5: Substitute values

Step 6: Calculate (show work!)

Step 7: Include correct units

Step 8: Check - does it make sense?

---

Composite 3D Figures

Sometimes shapes are combined!

Example: A building shaped like a rectangular prism with a square pyramid roof.

• Prism: 10 m × 10 m × 20 m
• Pyramid: base 10 m × 10 m, height 5 m

Total Volume: Prism: V₁ = 10 × 10 × 20 = 2,000 m³ Pyramid: V₂ = (1/3)(10²)(5) = 500/3 ≈ 166.67 m³ Total: 2,000 + 166.67 = 2,166.67 m³

Answer: About 2,167 cubic meters

---

When to Use Which Measurement?

Use Volume when:

• Finding capacity (how much it holds)
• Calculating amount of liquid, gas, or material
• Determining weight if you know density
• Example: How much water in a pool?

Use Surface Area when:

• Finding amount of material to cover/wrap
• Calculating paint needed
• Determining cost based on outside area
• Example: How much wrapping paper for a gift?

---

Real-World Careers Using These Skills

• Architecture: Designing buildings, calculating materials
• Engineering: Creating products, containers
• Manufacturing: Packaging design
• Construction: Estimating concrete, paint, materials
• Science: Lab measurements, experiments
• Cooking: Recipe scaling, container sizes

---

Summary

Volume measures space inside (cubic units)

• How much it can hold
• Prisms: Base Area × Height
• Pyramids/Cones: (1/3) × Base Area × Height
• Sphere: (4/3)πr³

Surface Area measures outside covering (square units)

• How much material to wrap it
• Add up all face areas
• Rectangular prism: 2(lw + lh + wh)
• Cylinder: 2πr² + 2πrh
• Sphere: 4πr²

Key Skills:

• Identify the 3D shape
• Use correct formula
• Keep units consistent
• Include proper units in answer

Remember:

• Volume = cubic units (cm³, in³, m³)
• Surface Area = square units (cm², in², m²)
• Radius = half the diameter
• Cone/Pyramid volume = 1/3 of prism/cylinder

Understanding volume and surface area is essential for architecture, engineering, science, and countless real-world applications!