Unit: Area Of Shapes
Chapter: Area of Cube, Cuboid & Cylinders
Reference: – What is Surface Area, Lateral Surface Area vs Total Surface Area, Cube – Definition, Properties, TSA and LSA Formulas, Cuboid – Definition, Properties, TSA and LSA Formulas, Cylinder – Definition, Properties, TSA and LSA Formulas, Real-Life Applications, Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- Difference Between Lateral and Total Surface Area
- How to Find Surface Area of a Cube
- How to Find Surface Area of a Cuboid
- How to Find Surface Area of a Cylinder
- When to Use Each Formula
Introduction to Surface Area of 3D Shapes
Definition
Surface area is the total area of all the faces (surfaces) of a three-dimensional object. It is measured in square units (cm², m², in², etc.). For cubes, cuboids, and cylinders, we can calculate both lateral surface area (area of only the curved or side faces) and total surface area (area of all faces including top and bottom).
When we calculate surface area, we essentially ask:
"How much material would it take to cover the entire outside of this 3D shape?"
Understanding surface area helps in painting, wrapping, packaging, and manufacturing.
Importance of Surface Area
- Used in packaging (amount of cardboard for a box)
- Used in painting (amount of paint needed for a wall or tank)
- Used in manufacturing (material required to make an object)
- Foundation for volume and other geometry concepts
Example
A cube with side 4 cm has a total surface area of 96 cm². A cuboid with dimensions 2 cm, 3 cm, 5 cm has total surface area 62 cm². A cylinder with radius 7 cm and height 10 cm has total surface area 748 cm².
Subtopics
1. Important Terms
Lateral Surface Area (LSA): The area of all the faces EXCEPT the top and bottom faces. For a cylinder, it is the curved surface area.
Total Surface Area (TSA): The area of ALL faces (including top and bottom). This is the sum of lateral surface area and the area of the two bases.
2. Cube
A cube is a 3D shape with 6 identical square faces. All edges are equal in length.
Properties of a Cube:
- 6 faces (all squares)
- 12 edges (all equal)
- 8 vertices
Lateral Surface Area of Cube: LSA = 4s²
(Only the 4 side faces, excluding top and bottom)
Total Surface Area of Cube: TSA = 6s²
(All 6 faces)
Where s = length of one edge (side)
Example: Cube with side 5 cm
LSA = 4 × 5² = 4 × 25 = 100 cm²
TSA = 6 × 5² = 6 × 25 = 150 cm²
3. Cuboid
A cuboid is a 3D shape with 6 rectangular faces. Opposite faces are identical. It has length (l), width (w), and height (h).
Properties of a Cuboid:
- 6 faces (all rectangles)
- 12 edges
- 8 vertices
Lateral Surface Area of Cuboid: LSA = 2h(l + w)
(Area of the 4 side faces, excluding top and bottom)
Total Surface Area of Cuboid: TSA = 2(lw + lh + wh)
(All 6 faces: front/back, left/right, top/bottom)
Example: Cuboid with l = 6 cm, w = 4 cm, h = 3 cm
LSA = 2 × 3 × (6 + 4) = 6 × 10 = 60 cm²
TSA = 2[(6×4) + (6×3) + (4×3)] = 2(24 + 18 + 12) = 2 × 54 = 108 cm²
4. Cylinder
A cylinder is a 3D shape with two parallel circular bases and one curved surface.
Properties of a Cylinder:
- 2 circular bases (top and bottom)
- 1 curved lateral surface
- No vertices
Lateral Surface Area (Curved Surface Area) of Cylinder: LSA = 2πrh
(Area of the curved surface only)
Total Surface Area of Cylinder: TSA = 2πr(r + h) OR TSA = 2πr² + 2πrh
(Area of curved surface + area of two circular bases)
Where r = radius of the base, h = height of the cylinder
Example: Cylinder with r = 7 cm, h = 10 cm
LSA = 2 × (22/7) × 7 × 10 = 2 × 22 × 10 = 440 cm² (using π = 22/7)
TSA = 2 × (22/7) × 7 × (7 + 10) = 2 × 22 × 17 = 748 cm²
Solved Examples
Example 1 – Cube TSA: Find the total surface area of a cube with side 7 cm.
Solution: TSA = 6s² = 6 × 7² = 6 × 49 = 294 cm²
Answer: 294 cm²
Example 2 – Cube LSA: Find the lateral surface area of a cube with side 10 cm.
Solution: LSA = 4s² = 4 × 10² = 4 × 100 = 400 cm²
Answer: 400 cm²
Example 3 – Cuboid TSA: Find the total surface area of a cuboid with dimensions l = 8 cm, w = 5 cm, h = 4 cm.
Solution: TSA = 2(lw + lh + wh) = 2[(8×5) + (8×4) + (5×4)] = 2(40 + 32 + 20) = 2 × 92 = 184 cm²
Answer: 184 cm²
Example 4 – Cuboid LSA: Find the lateral surface area of a cuboid with l = 12 cm, w = 8 cm, h = 6 cm.
Solution: LSA = 2h(l + w) = 2 × 6 × (12 + 8) = 12 × 20 = 240 cm²
Answer: 240 cm²
Example 5 – Cylinder LSA: Find the curved surface area of a cylinder with radius 14 cm and height 20 cm. (Use π = 22/7)
Solution: LSA = 2πrh = 2 × (22/7) × 14 × 20 = 2 × 22 × 2 × 20 = 2 × 22 × 40 = 1760 cm²
Answer: 1760 cm²
Example 6 – Cylinder TSA: Find the total surface area of a cylinder with radius 7 cm and height 15 cm. (Use π = 22/7)
Solution: TSA = 2πr(r + h) = 2 × (22/7) × 7 × (7 + 15) = 2 × 22 × 22 = 968 cm²
Answer: 968 cm²
Common Mistakes to Avoid
Mistake 1 – Confusing LSA with TSA
Lateral surface area excludes top and bottom; total surface area includes them.
Correct understanding: Think about whether the problem asks for "all faces" or "only the sides."
Mistake 2 – Using wrong formula for cuboid LSA
LSA = 2h(l + w), NOT 2(lw + lh + wh) (that is TSA).
Correct understanding: Memorize both formulas separately.
Mistake 3 – Forgetting the 2 in cylinder TSA
TSA = 2πr² + 2πrh, not just πr² + 2πrh.
Correct understanding: There are two circular bases, so 2πr² for both.
Mistake 4 – Using diameter instead of radius
Cylinder formulas use radius (r), not diameter (d).
Correct understanding: If given diameter, divide by 2 to find radius.
Mistake 5 – Mixing up cube and cuboid formulas
Cube: TSA = 6s², LSA = 4s². Cuboid: different formulas.
Correct understanding: Cube is a special case of cuboid where l = w = h = s.
Mistake 6 – Forgetting square units
Area is always in square units (cm², m², in², etc.).
Correct understanding: Don't write just cm; add the ² (squared).
Quick Reference Summary
Cube (s = side):
LSA = 4s²
TSA = 6s²
Cuboid (l = length, w = width, h = height):
LSA = 2h(l + w)
TSA = 2(lw + lh + wh)
Cylinder (r = radius, h = height):
LSA (Curved) = 2πrh
TSA = 2πr(r + h) = 2πr² + 2πrh
Remember:
π is approximately 3.14 or 22/7
LSA excludes top and bottom
TSA includes all faces