Surface Area Introduction
Surface area is the total area of all the outer faces of a 3D (three-dimensional) solid. When you wrap a gift box in paper, the amount of paper you use covers the surface area of the box.
In Class 5, you will learn to find the surface area of cuboids (rectangular boxes) and cubes by finding the area of each face and adding them up. A cuboid has 6 faces, and a cube (special cuboid with all equal sides) also has 6 faces.
Surface area is different from volume: surface area measures the outside covering, while volume measures the inside space.
What is Surface Area Introduction - Class 5 Maths (Measurement)?
The surface area of a 3D shape is the sum of the areas of all its faces.
- A cuboid has 6 rectangular faces: top, bottom, front, back, left, right.
- A cube has 6 equal square faces.
Faces of a cuboid with length (l), breadth (b), and height (h):
- Top and bottom: l × b (2 faces)
- Front and back: l × h (2 faces)
- Left and right: b × h (2 faces)
Surface Area Introduction Formula
Surface Area of Cuboid = 2(lb + bh + lh)
Surface Area of Cube = 6 × side²
Where l = length, b = breadth, h = height.
Types and Properties
Types of surface area:
- Total surface area (TSA): The area of all 6 faces. Used when wrapping a box completely.
- Lateral surface area (LSA): The area of only the 4 side faces (excluding top and bottom). Used when painting the walls of a room.
Lateral Surface Area of Cuboid = 2h(l + b)
Solved Examples
Example 1: Example 1: Surface Area of a Cuboid
Problem: Find the surface area of a cuboid with length 8 cm, breadth 5 cm, and height 3 cm.
Solution:
Step 1: SA = 2(lb + bh + lh)
Step 2: = 2(8×5 + 5×3 + 8×3)
Step 3: = 2(40 + 15 + 24) = 2 × 79 = 158 cm²
Answer: The surface area is 158 cm².
Example 2: Example 2: Surface Area of a Cube
Problem: Find the surface area of a cube with side 6 cm.
Solution:
Step 1: SA = 6 × side² = 6 × 6² = 6 × 36 = 216 cm²
Answer: The surface area is 216 cm².
Example 3: Example 3: Wrapping a Gift Box
Problem: A gift box is 20 cm long, 15 cm wide, and 10 cm high. How much wrapping paper is needed to cover it completely?
Solution:
Step 1: SA = 2(20×15 + 15×10 + 20×10)
Step 2: = 2(300 + 150 + 200) = 2 × 650 = 1,300 cm²
Answer: 1,300 cm² of wrapping paper is needed.
Example 4: Example 4: Painting the Walls of a Room
Problem: Ria’s room is 5 m long, 4 m wide, and 3 m high. Find the area of the four walls (lateral surface area).
Solution:
Step 1: LSA = 2h(l + b) = 2 × 3 × (5 + 4)
Step 2: = 6 × 9 = 54 m²
Answer: The area of the four walls is 54 m².
Example 5: Example 5: Finding Area of Each Face
Problem: A cuboid has dimensions 12 cm × 8 cm × 5 cm. Find the area of each pair of faces.
Solution:
Top & bottom = 12 × 8 = 96 cm² each (total: 192 cm²)
Front & back = 12 × 5 = 60 cm² each (total: 120 cm²)
Left & right = 8 × 5 = 40 cm² each (total: 80 cm²)
Total SA = 192 + 120 + 80 = 392 cm²
Answer: The total surface area is 392 cm².
Example 6: Example 6: Cost of Painting a Box
Problem: A wooden box is 30 cm × 20 cm × 15 cm. Painting costs ₹2 per cm². Find the total cost of painting the box on all sides.
Solution:
Step 1: SA = 2(30×20 + 20×15 + 30×15) = 2(600 + 300 + 450) = 2 × 1,350 = 2,700 cm²
Step 2: Cost = 2,700 × 2 = ₹5,400
Answer: The total cost is ₹5,400.
Example 7: Example 7: Comparing Surface Areas
Problem: Cube A has side 5 cm. Cube B has side 10 cm. How many times greater is Cube B’s surface area?
Solution:
SA of A = 6 × 25 = 150 cm²
SA of B = 6 × 100 = 600 cm²
Ratio = 600 ÷ 150 = 4
Answer: Cube B’s surface area is 4 times greater.
Example 8: Example 8: Open Box (No Lid)
Problem: An open box (no top) has dimensions 10 cm × 8 cm × 6 cm. Find its surface area.
Solution:
Step 1: Full SA = 2(10×8 + 8×6 + 10×6) = 2(80 + 48 + 60) = 376 cm²
Step 2: Remove top: 376 − (10 × 8) = 376 − 80 = 296 cm²
Answer: The surface area of the open box is 296 cm².
Example 9: Example 9: Dice Surface Area
Problem: A dice is a cube with side 2 cm. Find its total surface area.
Solution:
SA = 6 × 2² = 6 × 4 = 24 cm²
Answer: The surface area of the dice is 24 cm².
Example 10: Example 10: Surface Area vs Volume
Problem: A cube has side 4 cm. Find both the surface area and the volume. Are they the same?
Solution:
SA = 6 × 16 = 96 cm²
Volume = 4 × 4 × 4 = 64 cm³
Answer: SA = 96 cm², Volume = 64 cm³. They are different. Surface area measures covering (cm²); volume measures space inside (cm³).
Real-World Applications
Where do we use surface area?
- Wrapping gifts: Calculating how much wrapping paper is needed.
- Painting: Finding the area of walls or boxes to be painted.
- Packaging: Companies calculate surface area to design boxes and labels.
- Construction: Estimating material for walls, floors, and ceilings.
- Science: Heat loss through a surface depends on surface area.
Key Points to Remember
- Surface area = total area of all faces of a 3D shape.
- Cuboid SA = 2(lb + bh + lh). Cube SA = 6 × side².
- Lateral surface area (4 walls only) = 2h(l + b).
- Surface area is measured in square units (cm², m²).
- Surface area and volume are different: SA measures covering, volume measures inside space.
- For an open box (no lid), subtract the area of the missing face from the total SA.
- A cube is a special cuboid where l = b = h.
Practice Problems
- Find the surface area of a cuboid with l = 10 cm, b = 6 cm, h = 4 cm.
- Find the surface area of a cube with side 9 cm.
- A box is 25 cm × 15 cm × 10 cm. How much cardboard is needed to make it?
- Find the lateral surface area of a room 6 m × 4 m × 3 m.
- An open box (no top) is 12 cm × 10 cm × 8 cm. Find its surface area.
- A cube has surface area 294 cm². Find the length of one side.
- Which has a greater surface area: a cube of side 7 cm or a cuboid of 8 cm × 6 cm × 5 cm?
- Painting a cuboid box costs ₹3 per cm². The box is 15 cm × 10 cm × 8 cm. Find the total painting cost.
Frequently Asked Questions
Q1. What is surface area?
Surface area is the total area of all the outer faces of a 3D shape. It measures how much material would be needed to cover the outside of the shape.
Q2. What is the difference between surface area and volume?
Surface area measures the outside covering of a shape (in cm²). Volume measures the space inside the shape (in cm³). They use different formulas and different units.
Q3. How many faces does a cuboid have?
A cuboid has 6 rectangular faces: top, bottom, front, back, left, and right. Opposite faces are equal.
Q4. What is lateral surface area?
Lateral surface area (LSA) is the area of only the side faces, excluding the top and bottom. For a cuboid: LSA = 2h(l + b).
Q5. How do you find the surface area of a cube?
A cube has 6 equal square faces. SA = 6 × side². For a cube of side 5 cm: SA = 6 × 25 = 150 cm².
Q6. What if the box has no lid?
For an open box, calculate the full surface area, then subtract the area of the missing face (usually the top = l × b).
Q7. How do you find the side of a cube from its surface area?
Divide the surface area by 6 to get the area of one face. Then find the square root. Side = √(SA/6).
Q8. What units are used for surface area?
Surface area is measured in square units: cm² (square centimetres), m² (square metres), etc.
Q9. Is surface area taught in NCERT Class 5?
Yes. Introduction to surface area of cubes and cuboids is part of the Measurement chapter in NCERT/CBSE Class 5 Maths.
