Surface Area and Volume

Surface area tells you how much material you need to cover a 3D shape!

WHAT IS SURFACE AREA? Surface area (SA) is the sum of the areas of ALL faces of a three-dimensional figure.

Measured in square units: cm², ft², m², in²

RECTANGULAR PRISM:

A rectangular prism has 6 rectangular faces (like a box).

Dimensions: - Length (l) - Width (w) - Height (h)

Formula: SA = 2lw + 2lh + 2wh

OR think of it as: SA = 2(lw + lh + wh)

Why this works: - Top and bottom faces: 2 × (l × w) - Front and back faces: 2 × (l × h) - Left and right faces: 2 × (w × h)

Example: Find SA of a box: length = 5 cm, width = 3 cm, height = 4 cm

SA = 2(5)(3) + 2(5)(4) + 2(3)(4) SA = 2(15) + 2(20) + 2(12) SA = 30 + 40 + 24 SA = 94 cm²

CUBE:

A cube is a special prism where all edges are equal (s = side length). A cube has 6 identical square faces.

Formula: SA = 6s²

Why? Each face has area s², and there are 6 faces.

Example: Find SA of a cube with side length 4 inches.

SA = 6(4)² SA = 6(16) SA = 96 in²

USING NETS:

A net is a 2D pattern that folds into a 3D shape.

To find SA using a net: 1. Draw or visualize the net (unfolded shape) 2. Find the area of each face 3. Add all the areas together

Example Net of a Cube (side = 3):
    [3×3]
[3×3][3×3][3×3][3×3]
    [3×3]

SA = 6 faces × (3 × 3) = 6 × 9 = 54 square units

TRIANGULAR PRISM:

Has 2 triangular bases and 3 rectangular faces.

Formula: SA = 2B + Ph

Where: - B = area of triangular base - P = perimeter of triangular base - h = height (length) of prism

Example: Triangular base: base = 6 cm, height = 4 cm Prism height: 10 cm Side lengths of triangle: 6, 5, 5 cm

B = (1/2)(6)(4) = 12 cm² P = 6 + 5 + 5 = 16 cm

SA = 2(12) + (16)(10) SA = 24 + 160 SA = 184 cm²

VOLUME vs SURFACE AREA:

Don't confuse them!

Volume: Space INSIDE (cubic units) - How much it holds - Units: cm³, ft³, m³

Surface Area: Area of OUTSIDE (square units) - How much covers it - Units: cm², ft², m²

VOLUME FORMULAS:

Rectangular Prism: V = l × w × h Cube: V = s³ Triangular Prism: V = (1/2) × b × h × length

REAL-WORLD APPLICATIONS:

Surface Area: - Paint needed for walls - Wrapping paper for a gift - Material for a tent - Cardboard for a box - Metal for a storage container

Volume: - Water in a pool - Capacity of a refrigerator - Concrete for a foundation - Grain in a silo - Liquid in a container

PROBLEM-SOLVING TIPS:

1. Identify the shape: Cube, rectangular prism, triangular prism? 2. List given information: Label l, w, h, or s 3. Choose the correct formula 4. Substitute values carefully 5. Calculate step-by-step 6. Include units (square units for SA, cubic units for V) 7. Check reasonableness: Is your answer sensible?

Calculate surface area and volume of 3D shapes!

SURFACE AREA - CUBES: 1. Find SA of a cube with side length 5 cm. 2. Find SA of a cube with edge length 7 inches. 3. A cube has side length 10 m. What is its surface area?

SURFACE AREA - RECTANGULAR PRISMS: 4. Find SA: length = 8 ft, width = 3 ft, height = 5 ft 5. Find SA: l = 12 cm, w = 4 cm, h = 6 cm 6. Find SA: l = 10 in, w = 10 in, h = 15 in

VOLUME - CUBES: 7. Find volume of a cube with side 4 cm. 8. Find volume of a cube with edge 9 inches.

VOLUME - RECTANGULAR PRISMS: 9. Find volume: l = 6 m, w = 4 m, h = 5 m 10. Find volume: l = 10 ft, w = 8 ft, h = 3 ft 11. Find volume: l = 7 cm, w = 7 cm, h = 12 cm

WORD PROBLEMS - SURFACE AREA: 12. A box is 15 inches long, 8 inches wide, and 10 inches tall. How much cardboard is needed to make the box?

13. You're painting a cube-shaped storage unit with 6-foot edges. One can of paint covers 50 ft². How many cans do you need?

14. A rectangular room is 12 ft long, 10 ft wide, and 8 ft high. You want to paint the walls and ceiling (not the floor). What is the surface area to paint?

WORD PROBLEMS - VOLUME: 15. An aquarium is 20 inches long, 10 inches wide, and 12 inches high. What is its volume?

16. A refrigerator has interior dimensions: 3 ft wide, 2.5 ft deep, 5 ft tall. What is its capacity in cubic feet?

17. How many cubic inches of water can a cube-shaped container with 8-inch edges hold?

MIXED PROBLEMS: 18. A cube has surface area 96 cm². What is the length of one edge? (Hint: Use SA = 6s²)

19. A rectangular prism has volume 240 cm³, length 8 cm, and width 5 cm. What is the height?

CHALLENGE:
20. A cube has volume 64 in³. What is its surface area?
    (Hint: Find the edge length first using V = s³)

21. A rectangular prism has dimensions 4 × 6 × 8 cm. If each dimension is doubled, how does the surface area change? How does volume change?

22. A triangular prism has triangular bases with base 6 cm and height 8 cm. The prism length is 15 cm. The triangle side lengths are 6, 8, and 10 cm. Find the surface area.