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Calculate surface area and volume of 3D shapes
Learn step-by-step with practice exercises built right in.
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!
2D shapes are flat (like squares, circles, triangles)
3D shapes have length, width, AND height
Common 3D shapes:
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)
Find the volume of a rectangular prism with length 6 cm, width 4 cm, and height 5 cm.
Use the formula: V = lwh
V = 6 × 4 × 5 V = 120 cm³
Answer: 120 cm³
Find the surface area of a cube with side length 3 inches.
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Example: A box that is 3 cm × 4 cm × 2 cm has volume = 24 cm³ (24 cubic centimeters of space inside)
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
A box with length (l), width (w), and height (h).
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
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
A cube is a special rectangular prism where all edges are equal (s = side length).
V = s³ (side × side × side)
Example: A cube with side = 6 cm
Solution: V = 6³ = 6 × 6 × 6 = 216 cm³
Answer: 216 cubic centimeters
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!
A cylinder has two circular bases and a curved side.
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 = 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
A prism with triangular bases.
V = (Area of base) × height
For triangular base: V = (1/2 × base × height of triangle) × prism height
Example: Triangular prism with:
Solution: V = (1/2 × 6 × 4) × 10 V = (12) × 10 V = 120 cm³
Answer: 120 cubic centimeters
Add: 2 triangular bases + 3 rectangular sides
Example: Calculate each face area and add them all up!
A cone has one circular base and comes to a point (apex).
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!
A perfectly round ball shape.
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
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
A pyramid has a base (often square) and triangular faces meeting at an apex.
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!
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
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
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)
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
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
Notice the patterns:
Prisms: V = (Base Area) × Height
Pyramids/Cones: V = (1/3) × (Base Area) × Height
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)
Sometimes you know the volume and need to find a dimension!
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
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
Important: Keep units consistent!
Volume units:
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
❌ Mistake 1: Confusing volume and surface area
❌ Mistake 2: Forgetting the 1/3 for cones and pyramids
❌ Mistake 3: Using diameter instead of radius
❌ Mistake 4: Wrong units
❌ Mistake 5: Calculation errors with π
❌ Mistake 6: Mixing up dimensions
Tip 1: Draw and label
Tip 2: Identify the shape
Tip 3: Check what's given
Tip 4: Use formulas correctly
Tip 5: Check reasonableness
Tip 6: Remember special cases
Rectangular Prism:
Cube:
Cylinder:
Cone:
Sphere:
Square Pyramid:
Triangular Prism:
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?
Sometimes shapes are combined!
Example: A building shaped like a rectangular prism with a square pyramid roof.
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
Use Volume when:
Use Surface Area when:
Volume measures space inside (cubic units)
Surface Area measures outside covering (square units)
Key Skills:
Remember:
Understanding volume and surface area is essential for architecture, engineering, science, and countless real-world applications!
Use the formula: SA = 6s²
SA = 6 × (3²) SA = 6 × 9 SA = 54 in²
Answer: 54 in²
A cylinder has a radius of 4 cm and height of 10 cm. Find the volume. Use π ≈ 3.14.
Use the formula: V = πr²h
V = 3.14 × (4²) × 10 V = 3.14 × 16 × 10 V = 3.14 × 160 V = 502.4 cm³
Answer: 502.4 cm³ (or about 502 cm³)
A rectangular box is 8 ft long, 5 ft wide, and 3 ft tall. How much cardboard is needed to make the box (surface area)?
Use the formula: SA = 2(lw + lh + wh)
SA = 2(8×5 + 8×3 + 5×3) SA = 2(40 + 24 + 15) SA = 2(79) SA = 158 ft²
Answer: 158 ft²
A cone-shaped container has a radius of 6 inches and a height of 15 inches. How much liquid can it hold? Use π ≈ 3.14.
Use the formula: V = (1/3)πr²h
V = (1/3) × 3.14 × (6²) × 15 V = (1/3) × 3.14 × 36 × 15 V = (1/3) × 3.14 × 540 V = (1/3) × 1,695.6 V = 565.2 in³
Answer: 565.2 in³ (or about 565 cubic inches)