Surface Area of Cuboid
A cuboid is a three-dimensional solid shape with six rectangular faces, eight vertices, and twelve edges. It is one of the most common shapes found in everyday life — books, bricks, boxes, and rooms are all cuboids.
The surface area of a cuboid is the total area of all its six rectangular faces. When you wrap a gift box or paint the walls of a room, you are working with the surface area of a cuboid.
In Class 8 Mathematics (NCERT), the surface area of cuboid is studied under the chapter Mensuration. You need to know the three dimensions of the cuboid — length (l), breadth (b), and height (h) — to calculate its surface area.
There are two types of surface area for a cuboid: Total Surface Area (TSA), which includes all six faces, and Lateral Surface Area (LSA), which includes only the four side faces (excluding the top and bottom).
What is Surface Area of Cuboid?
Definition: The surface area of a cuboid is the sum of the areas of all six rectangular faces.
Key terms:
- Cuboid: A 3D shape with 6 rectangular faces, 8 vertices, and 12 edges.
- Length (l): The longest dimension of the cuboid.
- Breadth (b): The dimension perpendicular to the length (width).
- Height (h): The vertical dimension of the cuboid.
- Total Surface Area (TSA): The area of ALL six faces.
- Lateral Surface Area (LSA): The area of only the four side faces (walls), excluding the top and bottom.
Understanding the six faces:
- A cuboid has 3 pairs of opposite, identical rectangular faces.
- Top and Bottom: Each has area = l × b. (2 faces)
- Front and Back: Each has area = l × h. (2 faces)
- Left and Right: Each has area = b × h. (2 faces)
Surface Area of Cuboid Formula
Total Surface Area (TSA) of Cuboid:
TSA = 2(lb + bh + hl)
Where:
- l = length of the cuboid
- b = breadth of the cuboid
- h = height of the cuboid
Lateral Surface Area (LSA) of Cuboid:
LSA = 2h(l + b)
Where:
- LSA = area of the four side faces only (no top, no bottom).
- Also written as: LSA = 2(lh + bh) = 2h(l + b).
Relationship:
- TSA = LSA + 2(l × b)
- TSA = LSA + area of top + area of bottom
Derivation and Proof
Deriving the formula step by step:
A cuboid has 6 faces in 3 pairs:
- Top and Bottom faces: Each is a rectangle with dimensions l × b. Area of one face = lb. Total = 2 × lb = 2lb.
- Front and Back faces: Each is a rectangle with dimensions l × h. Area of one face = lh. Total = 2 × lh = 2lh.
- Left and Right faces: Each is a rectangle with dimensions b × h. Area of one face = bh. Total = 2 × bh = 2bh.
- TSA = 2lb + 2lh + 2bh
- TSA = 2(lb + bh + hl)
- LSA = Front + Back + Left + Right
- LSA = 2lh + 2bh
- LSA = 2h(l + b)
Alternate understanding: If you open a cuboid like a cardboard box and lay all faces flat, you get a net. The total area of the net equals the TSA of the cuboid.
Special case — Cube:
- When l = b = h = a (all sides equal), the cuboid becomes a cube.
- TSA of cube = 2(a×a + a×a + a×a) = 2(3a²) = 6a².
- LSA of cube = 2a(a + a) = 4a².
Types and Properties
Types of surface area problems:
1. Finding TSA given dimensions:
- Directly apply TSA = 2(lb + bh + hl).
- Most straightforward type of problem.
2. Finding LSA given dimensions:
- Apply LSA = 2h(l + b).
- Used in problems about painting walls, fencing, etc.
3. Finding a missing dimension given TSA:
- Substitute known values into the formula and solve for the unknown.
- Example: Given TSA, l, and b, find h.
4. Painting/wrapping problems:
- To paint the four walls of a room: Use LSA = 2h(l + b).
- To wrap a box completely: Use TSA = 2(lb + bh + hl).
- Remember to account for doors and windows in room-painting problems (subtract their area).
5. Cost-based problems:
- Find the surface area first, then multiply by the cost per unit area.
- Example: Cost of painting = LSA × cost per m².
6. Open-top box problems:
- A box without a lid has only 5 faces.
- Surface area = TSA − area of top = 2(lb + bh + hl) − lb = lb + 2bh + 2hl.
Solved Examples
Example 1: Example 1: Basic TSA calculation
Problem: Find the total surface area of a cuboid with length = 10 cm, breadth = 6 cm, and height = 4 cm.
Solution:
Given:
- l = 10 cm, b = 6 cm, h = 4 cm
Using the formula:
- TSA = 2(lb + bh + hl)
- TSA = 2(10×6 + 6×4 + 4×10)
- TSA = 2(60 + 24 + 40)
- TSA = 2(124)
- TSA = 248 cm²
Answer: The total surface area is 248 cm².
Example 2: Example 2: LSA calculation
Problem: Find the lateral surface area of a cuboid with dimensions 12 cm × 8 cm × 5 cm.
Solution:
Given:
- l = 12 cm, b = 8 cm, h = 5 cm
Using the formula:
- LSA = 2h(l + b)
- LSA = 2 × 5 × (12 + 8)
- LSA = 10 × 20
- LSA = 200 cm²
Answer: The lateral surface area is 200 cm².
Example 3: Example 3: Painting the walls of a room
Problem: A room is 15 m long, 10 m wide, and 4 m high. Find the cost of painting its four walls at Rs 25 per m².
Solution:
Given:
- l = 15 m, b = 10 m, h = 4 m, Rate = Rs 25/m²
Step 1: Find LSA (four walls):
- LSA = 2h(l + b) = 2 × 4 × (15 + 10) = 8 × 25 = 200 m²
Step 2: Calculate cost:
- Cost = 200 × 25 = Rs 5,000
Answer: The cost of painting the four walls is Rs 5,000.
Example 4: Example 4: Finding a missing dimension
Problem: The TSA of a cuboid is 340 cm². If its length is 10 cm and breadth is 8 cm, find its height.
Solution:
Given:
- TSA = 340 cm², l = 10 cm, b = 8 cm, h = ?
Using the formula:
- 340 = 2(10×8 + 8×h + h×10)
- 340 = 2(80 + 8h + 10h)
- 340 = 2(80 + 18h)
- 170 = 80 + 18h
- 18h = 170 − 80 = 90
- h = 90/18 = 5 cm
Answer: The height is 5 cm.
Example 5: Example 5: Wrapping a box
Problem: How much wrapping paper is needed to cover a box 25 cm long, 15 cm wide, and 10 cm tall? (No overlap.)
Solution:
Given:
- l = 25 cm, b = 15 cm, h = 10 cm
Paper needed = TSA of the box:
- TSA = 2(lb + bh + hl)
- TSA = 2(25×15 + 15×10 + 10×25)
- TSA = 2(375 + 150 + 250)
- TSA = 2(775)
- TSA = 1,550 cm²
Answer: 1,550 cm² of wrapping paper is needed.
Example 6: Example 6: Open-top box
Problem: Find the surface area of an open-top box with length 20 cm, breadth 12 cm, and height 8 cm.
Solution:
Given:
- l = 20 cm, b = 12 cm, h = 8 cm (no lid)
Surface area of open box = TSA − area of top:
- TSA = 2(20×12 + 12×8 + 8×20) = 2(240 + 96 + 160) = 2(496) = 992 cm²
- Area of top = l × b = 20 × 12 = 240 cm²
- Surface area of open box = 992 − 240 = 752 cm²
Answer: The surface area of the open-top box is 752 cm².
Example 7: Example 7: Room painting with door and window
Problem: A room is 12 m × 8 m × 3.5 m. It has a door (2 m × 1.5 m) and 2 windows (1.5 m × 1 m each). Find the area to be painted.
Solution:
Given:
- l = 12 m, b = 8 m, h = 3.5 m
Step 1: LSA of four walls:
- LSA = 2 × 3.5 × (12 + 8) = 7 × 20 = 140 m²
Step 2: Subtract door and windows:
- Door area = 2 × 1.5 = 3 m²
- Window area = 2 × (1.5 × 1) = 3 m²
- Total to subtract = 3 + 3 = 6 m²
Step 3: Area to paint = 140 − 6 = 134 m²
Answer: The area to be painted is 134 m².
Example 8: Example 8: Comparing two cuboids
Problem: Which has more surface area: Cuboid A (8 cm × 6 cm × 4 cm) or Cuboid B (10 cm × 5 cm × 3 cm)?
Solution:
Cuboid A:
- TSA = 2(8×6 + 6×4 + 4×8) = 2(48 + 24 + 32) = 2(104) = 208 cm²
Cuboid B:
- TSA = 2(10×5 + 5×3 + 3×10) = 2(50 + 15 + 30) = 2(95) = 190 cm²
Answer: Cuboid A has more surface area (208 cm² vs 190 cm²).
Example 9: Example 9: Converting units
Problem: Find the TSA of a cuboid with length = 1.5 m, breadth = 80 cm, and height = 60 cm. Give the answer in cm².
Solution:
Step 1: Convert all to cm:
- l = 1.5 m = 150 cm, b = 80 cm, h = 60 cm
Step 2: Apply formula:
- TSA = 2(150×80 + 80×60 + 60×150)
- TSA = 2(12,000 + 4,800 + 9,000)
- TSA = 2(25,800)
- TSA = 51,600 cm²
Answer: The TSA is 51,600 cm² or 5.16 m².
Example 10: Example 10: Cardboard for a box
Problem: A cardboard box (without a lid) is 30 cm long, 20 cm wide, and 15 cm deep. Find the area of cardboard needed and the cost at Rs 2 per cm².
Solution:
Given:
- l = 30 cm, b = 20 cm, h = 15 cm (no lid)
Step 1: Surface area of open box:
- TSA = 2(30×20 + 20×15 + 15×30) = 2(600 + 300 + 450) = 2(1350) = 2,700 cm²
- Subtract top: 2,700 − (30 × 20) = 2,700 − 600 = 2,100 cm²
Step 2: Cost = 2,100 × 2 = Rs 4,200
Answer: Cardboard needed = 2,100 cm². Cost = Rs 4,200.
Real-World Applications
Real-world applications of surface area of cuboid:
- Packaging: Finding the amount of material (cardboard, wrapping paper) needed to make or wrap a box.
- Painting rooms: Calculating the area of walls, ceiling, or floor to find the amount of paint required.
- Tiling: Finding the number of tiles needed for a floor (area = l × b) or walls (LSA).
- Heating and cooling: The surface area of a room determines heat loss/gain, which helps in sizing air conditioners and heaters.
- Manufacturing: Calculating material needed for metal boxes, containers, and cabinets.
- Construction: Estimating plaster, paint, or wallpaper for rooms and buildings.
- Shipping: Surface area determines the amount of material needed for packing cases and cartons.
Key Points to Remember
- A cuboid has 6 rectangular faces, 8 vertices, and 12 edges.
- TSA = 2(lb + bh + hl), where l = length, b = breadth, h = height.
- LSA = 2h(l + b) — area of the four side faces only.
- TSA = LSA + 2(l × b).
- For an open-top box: Surface area = TSA − l × b.
- All dimensions must be in the same unit before applying the formula.
- The unit of surface area is square units (cm², m², etc.).
- A cube is a special cuboid where l = b = h. TSA of cube = 6a².
- For room painting, use LSA and subtract areas of doors and windows.
- Surface area measures the total outer covering of a 3D shape.
Practice Problems
- Find the TSA of a cuboid with l = 14 cm, b = 10 cm, h = 6 cm.
- Find the LSA of a cuboid with dimensions 20 cm × 15 cm × 8 cm.
- A room is 18 m × 12 m × 4 m. Find the cost of painting four walls at Rs 30/m².
- The TSA of a cuboid is 430 cm². If l = 12 cm and b = 5 cm, find h.
- How much cardboard is needed to make an open-top box of dimensions 40 cm × 25 cm × 20 cm?
- A cuboid tank is 3 m × 2 m × 1.5 m. Find the area of sheet metal needed to make it (closed tank).
- Compare the TSA of a cuboid (6 × 5 × 4 cm) with a cube of side 5 cm.
- A room has 2 doors (2 m × 1 m each) and 4 windows (1 m × 0.5 m each). Room is 10 m × 8 m × 3 m. Find the area to be whitewashed.
Frequently Asked Questions
Q1. What is the surface area of a cuboid?
The surface area of a cuboid is the total area of all six rectangular faces. The formula is TSA = 2(lb + bh + hl), where l, b, h are the length, breadth, and height.
Q2. What is the difference between TSA and LSA?
TSA (Total Surface Area) includes all 6 faces. LSA (Lateral Surface Area) includes only the 4 side faces, excluding the top and bottom. TSA = LSA + 2(l × b).
Q3. What is the LSA formula for a cuboid?
LSA = 2h(l + b), which is the combined area of the four vertical faces (front, back, left, right).
Q4. When do we use LSA instead of TSA?
Use LSA when you need to find the area of only the side surfaces — for example, painting the walls of a room (not the ceiling or floor).
Q5. How is a cuboid different from a cube?
A cube has all sides equal (l = b = h = a), while a cuboid can have three different dimensions. Every cube is a cuboid, but not every cuboid is a cube.
Q6. What is the surface area of an open box?
For a box without a lid (open top), subtract the area of the top face from the TSA. Surface area = 2(lb + bh + hl) − lb = lb + 2bh + 2hl.
Q7. What units are used for surface area?
Surface area is measured in square units — cm², m², mm², etc. Make sure all dimensions are in the same unit before calculating.
Q8. How do you find the cost of painting a room?
Calculate the LSA of the room using 2h(l + b). Subtract the areas of doors and windows. Multiply the net area by the cost per square metre.
Q9. How many faces does a cuboid have?
A cuboid has 6 rectangular faces arranged in 3 pairs of identical opposite faces: top-bottom (l × b), front-back (l × h), and left-right (b × h).
Q10. Can the surface area formula be used for irregular boxes?
No. The formula TSA = 2(lb + bh + hl) applies only to rectangular cuboids. Irregular shapes require calculating the area of each face separately and adding them.
