Lateral Surface Area
The lateral surface area (LSA) of a solid is the total area of all its side faces, excluding the top and bottom faces (bases). It is also called the curved surface area (CSA) for solids with curved surfaces like cylinders and cones.
Understanding LSA is important when you need to find the area of the “walls” of a solid — for instance, the area to be painted on a room’s four walls (excluding floor and ceiling) or the label area on a tin can.
In Class 8, you study the lateral surface area of three common solids: cube, cuboid, and cylinder. Each has a distinct formula based on its shape and dimensions.
The concept of LSA vs. TSA (Total Surface Area) is a key distinction in mensuration. TSA includes the base areas, while LSA does not.
What is Lateral Surface Area?
Definition: The lateral surface area of a solid is the sum of the areas of all its faces except the top and bottom bases.
Key distinction:
- Lateral Surface Area (LSA) = area of side faces only
- Total Surface Area (TSA) = LSA + area of both bases
- For a cylinder, LSA is called Curved Surface Area (CSA)
When to use LSA:
- Painting the four walls of a room (not floor/ceiling)
- Wrapping a label around a can (not the circular ends)
- Finding the cost of material for the lateral surface of a chimney, pillar, or tank
Lateral Surface Area Formula
Lateral Surface Area of a Cube:
LSA = 4a²
Where a = side of the cube.
Lateral Surface Area of a Cuboid:
LSA = 2h(l + b)
Where l = length, b = breadth, h = height.
Curved Surface Area of a Cylinder:
CSA = 2πrh
Where r = radius and h = height of the cylinder.
Corresponding Total Surface Areas:
- Cube: TSA = 6a²
- Cuboid: TSA = 2(lb + bh + lh)
- Cylinder: TSA = 2πr(r + h)
Derivation and Proof
Derivation for Cuboid:
Step 1: A cuboid has 6 faces: top, bottom, front, back, left, and right.
Step 2: The lateral faces (excluding top and bottom) are:
- Front face: l × h
- Back face: l × h
- Left face: b × h
- Right face: b × h
Step 3: LSA = lh + lh + bh + bh = 2lh + 2bh = 2h(l + b)
Derivation for Cube:
A cube has l = b = h = a.
- LSA = 2a(a + a) = 2a(2a) = 4a²
Derivation for Cylinder:
Step 1: If you “unroll” the curved surface of a cylinder, you get a rectangle.
Step 2: The width of this rectangle equals the height h of the cylinder.
Step 3: The length of this rectangle equals the circumference of the circular base = 2πr.
Step 4: Area of the rectangle = length × width = 2πr × h = 2πrh.
Types and Properties
Problems on lateral surface area can be classified as follows:
1. Direct computation:
- Given dimensions, find the LSA using the appropriate formula.
2. Finding a missing dimension:
- Given LSA and some dimensions, find the unknown dimension.
3. Cost of painting/covering:
- Find LSA, then multiply by cost per unit area.
4. LSA vs. TSA comparison:
- Calculate both and find the difference (= area of bases).
5. Open-top containers:
- TSA of an open-top box = LSA + area of one base.
6. Combined shapes:
- A cylinder on top of a cuboid, or stacked solids — calculate exposed lateral surfaces.
Solved Examples
Example 1: Example 1: LSA of a cube
Problem: Find the lateral surface area of a cube with side 7 cm.
Solution:
- LSA = 4a² = 4 × 7² = 4 × 49 = 196 cm²
Answer: The LSA is 196 cm².
Example 2: Example 2: LSA of a cuboid
Problem: Find the lateral surface area of a cuboid with length 12 cm, breadth 8 cm, and height 5 cm.
Solution:
- LSA = 2h(l + b) = 2 × 5 × (12 + 8) = 10 × 20 = 200 cm²
Answer: The LSA is 200 cm².
Example 3: Example 3: CSA of a cylinder
Problem: Find the curved surface area of a cylinder with radius 7 cm and height 10 cm. (Use π = 22/7)
Solution:
- CSA = 2πrh = 2 × 22/7 × 7 × 10
- = 2 × 22 × 10 = 440 cm²
Answer: The CSA is 440 cm².
Example 4: Example 4: Painting walls of a room
Problem: A room is 6 m long, 4 m wide, and 3.5 m high. Find the cost of painting its four walls at ₹25 per m².
Solution:
- LSA = 2h(l + b) = 2 × 3.5 × (6 + 4) = 7 × 10 = 70 m²
- Cost = 70 × 25 = ₹1,750
Answer: The cost of painting is ₹1,750.
Example 5: Example 5: Finding height from CSA
Problem: The curved surface area of a cylinder is 1,320 cm² and its radius is 21 cm. Find the height. (Use π = 22/7)
Solution:
- CSA = 2πrh
- 1,320 = 2 × 22/7 × 21 × h
- 1,320 = 2 × 22 × 3 × h = 132h
- h = 1,320 / 132 = 10 cm
Answer: The height is 10 cm.
Example 6: Example 6: Label on a tin can
Problem: A cylindrical tin can has radius 3.5 cm and height 12 cm. Find the area of the paper label that wraps around it. (Use π = 22/7)
Solution:
- Label area = CSA = 2πrh
- = 2 × 22/7 × 3.5 × 12
- = 2 × 22/7 × 7/2 × 12
- = 2 × 22 × 6 = 264 cm²
Answer: The label area is 264 cm².
Example 7: Example 7: Difference between TSA and LSA of a cuboid
Problem: A cuboid has dimensions 10 cm × 6 cm × 4 cm. Find the difference between its TSA and LSA.
Solution:
- TSA = 2(lb + bh + lh) = 2(60 + 24 + 40) = 2 × 124 = 248 cm²
- LSA = 2h(l + b) = 2 × 4 × 16 = 128 cm²
- Difference = 248 − 128 = 120 cm²
Note: The difference = 2 × l × b = 2 × 10 × 6 = 120 cm² (area of top and bottom).
Answer: The difference is 120 cm².
Example 8: Example 8: Open-top box
Problem: An open-top box (no lid) has dimensions 30 cm × 20 cm × 15 cm. Find the total area of material needed.
Solution:
- LSA = 2 × 15 × (30 + 20) = 30 × 50 = 1,500 cm²
- Base area = 30 × 20 = 600 cm²
- Total = 1,500 + 600 = 2,100 cm²
Answer: The material needed is 2,100 cm².
Example 9: Example 9: Finding radius from CSA and height
Problem: A cylinder has CSA = 880 cm² and height = 20 cm. Find the radius. (Use π = 22/7)
Solution:
- 880 = 2 × 22/7 × r × 20
- 880 = 880r/7
- r = 880 × 7 / 880 = 7 cm
Answer: The radius is 7 cm.
Example 10: Example 10: Comparing LSA of cube and cuboid
Problem: A cube has side 6 cm. A cuboid has dimensions 8 cm × 6 cm × 4 cm. Which has a greater LSA?
Solution:
- LSA of cube = 4 × 6² = 4 × 36 = 144 cm²
- LSA of cuboid = 2 × 4 × (8 + 6) = 8 × 14 = 112 cm²
Answer: The cube has the greater LSA (144 cm² > 112 cm²).
Real-World Applications
Painting and Wallpapering: To find how much paint or wallpaper is needed for the walls of a room, calculate the LSA of the room (a cuboid) and subtract window/door areas.
Labelling: Manufacturers calculate the CSA of cylindrical cans and bottles to determine the size of paper labels.
Fencing and Cladding: The lateral surface of pillars, columns, and chimneys is calculated for cladding or plastering.
Heating and Cooling: Heat loss through walls depends on the LSA. Engineers calculate it to design insulation systems.
Packaging: The material for wrapping a box (excluding top/bottom) requires the LSA calculation.
Key Points to Remember
- LSA = area of side faces only, excluding top and bottom.
- TSA = LSA + area of both bases.
- Cube: LSA = 4a², TSA = 6a².
- Cuboid: LSA = 2h(l + b), TSA = 2(lb + bh + lh).
- Cylinder: CSA = 2πrh, TSA = 2πr(r + h).
- Difference between TSA and LSA = area of the two bases.
- For an open-top container, surface area = LSA + one base area.
- The curved surface of a cylinder, when unrolled, forms a rectangle of dimensions 2πr × h.
- LSA is measured in square units.
- Always check whether a problem asks for LSA or TSA before solving.
Practice Problems
- Find the LSA of a cube with side 9 cm.
- Find the LSA of a cuboid with l = 15 cm, b = 10 cm, h = 8 cm.
- Find the CSA of a cylinder with r = 14 cm and h = 20 cm. (Use π = 22/7)
- A room is 8 m × 5 m × 4 m. Find the cost of whitewashing its walls at ₹30 per m².
- The CSA of a cylinder is 2,200 cm² and its height is 25 cm. Find the radius.
- A cube has TSA 486 cm². Find its LSA.
- An open-top cylindrical tank has r = 7 m and h = 5 m. Find the total area of sheet metal used.
- Find the height of a cuboid whose LSA is 240 cm², length is 10 cm, and breadth is 6 cm.
Frequently Asked Questions
Q1. What is lateral surface area?
Lateral surface area is the total area of all the side faces of a solid, excluding the top and bottom faces (bases).
Q2. What is the difference between LSA and TSA?
LSA includes only the side faces. TSA includes the side faces plus both bases. So TSA = LSA + area of two bases.
Q3. Why is the lateral surface of a cylinder called curved surface area?
Because the side surface of a cylinder is curved, not flat. When unrolled, it forms a rectangle of dimensions 2πr × h.
Q4. How do you find the area to paint the four walls of a room?
Use LSA = 2h(l + b), where l is length, b is breadth, and h is height of the room. Subtract the area of doors and windows if specified.
Q5. What is the LSA of a cube with side a?
LSA = 4a². A cube has 4 lateral faces, each a square of area a².
Q6. Can LSA be greater than TSA?
No. TSA = LSA + area of bases, so TSA is always greater than or equal to LSA.
Q7. What is the LSA formula for a cylinder?
CSA = 2πrh, where r is the radius and h is the height.
Q8. How do you calculate the surface area of an open box?
Surface area = LSA + area of one base (since the top is open). For a cuboid: 2h(l+b) + lb.
Q9. What units are used for surface area?
Square units: cm², m², mm², etc. Never cubic units (those are for volume).
Q10. If the radius of a cylinder is doubled, how does CSA change?
CSA = 2πrh. If r is doubled, CSA becomes 2π(2r)h = 4πrh, which is double the original CSA.
