Surface Area and Volume
Explore the 3D world! Surface area measures the total area of all faces of a solid figure - imagine wrapping it with paper. Master surface area calculations for cubes and prisms to solve real-world packaging and construction problems.
Concept
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?
Try it
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.