Perimeter and Area – Class 7 CBSE Mathematics

Introduction

Perimeter and Area are fundamental concepts in geometry. Perimeter refers to the total length of the boundary of a two-dimensional shape, while Area refers to the amount of space enclosed within that boundary. This chapter introduces students to calculating the perimeter and area of basic shapes like squares, rectangles, triangles, and circles, along with real-life applications.

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1. Perimeter

The perimeter of a shape is the total distance around its outer boundary. It is measured in units like centimeters (cm), meters (m), etc.

Perimeter of a Square

• A square has four equal sides.
• Formula: Perimeter = 4 × side (P = 4s)
• Example: If the side of a square is 5 cm,
  Perimeter = 4 × 5 = 20 cm.

Perimeter of a Rectangle

• A rectangle has two pairs of equal sides: length (l) and breadth (b).
• Formula: Perimeter = 2 × (length + breadth) (P = 2(l + b))
• Example: If length = 6 cm and breadth = 4 cm,
  Perimeter = 2 × (6 + 4) = 2 × 10 = 20 cm.

Perimeter of a Triangle

• A triangle has three sides (say, a, b, and c).
• Formula: Perimeter = a + b + c
• Example: If sides are 3 cm, 4 cm, and 5 cm,
  Perimeter = 3 + 4 + 5 = 12 cm.

Real-Life Application

• Perimeter is used to calculate the length of fencing needed for a garden or a field.

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2. Area

The area measures the space inside a shape and is expressed in square units (e.g., cm², m²).

Area of a Square

• Formula: Area = side × side (A = s²)
• Example: If the side is 5 cm,
  Area = 5 × 5 = 25 cm².

Area of a Rectangle

• Formula: Area = length × breadth (A = l × b)
• Example: If length = 6 cm and breadth = 4 cm,
  Area = 6 × 4 = 24 cm².

Area of a Triangle

• Formula: Area = ½ × base × height (A = ½ × b × h)
  • Base (b) is the length of the bottom side.
  • Height (h) is the perpendicular distance from the base to the opposite vertex.
• Example: If base = 6 cm and height = 4 cm,
  Area = ½ × 6 × 4 = ½ × 24 = 12 cm².

Real-Life Application

• Area is used to determine the amount of paint needed for a wall or tiles for a floor.

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3. Special Cases

Area of a Parallelogram

• A parallelogram is a four-sided shape with opposite sides equal and parallel.
• Formula: Area = base × height (A = b × h)
  • Base is the length of one side.
  • Height is the perpendicular distance between the base and the opposite side.
• Example: If base = 5 cm and height = 3 cm,
  Area = 5 × 3 = 15 cm².

Difference Between Triangle and Parallelogram

• The area of a triangle is half that of a parallelogram with the same base and height because a triangle is essentially half of a parallelogram.

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4. Circles

A circle is a round shape with a single continuous boundary.

Circumference of a Circle (Perimeter)

• The perimeter of a circle is called its circumference.
• Formula: Circumference = 2 × π × radius (C = 2πr)
  • π (pi) ≈ 3.14 or 22/7.
  • Radius (r) is the distance from the center to the boundary.
• Example: If radius = 7 cm,
  Circumference = 2 × 22/7 × 7 = 44 cm.

Area of a Circle

• Formula: Area = π × radius × radius (A = πr²)
• Example: If radius = 7 cm,
  Area = 22/7 × 7 × 7 = 22/7 × 49 = 154 cm².

Key Points

• Use 22/7 for π when the radius is a multiple of 7 for easier calculations; otherwise, use 3.14.

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5. Conversion of Units

When solving problems, ensure units are consistent.

• Length: 1 m = 100 cm, 1 cm = 10 mm.
• Area: 1 m² = 10,000 cm² (100 × 100), 1 cm² = 100 mm².
• Example: If length = 2 m and breadth = 50 cm,
  Convert 2 m to 200 cm,
  Area = 200 × 50 = 10,000 cm² = 1 m².

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6. Applications of Perimeter and Area

Word Problems

1. Fencing a Garden: A rectangular garden is 12 m long and 8 m wide. How much fencing is needed?
   • Perimeter = 2 × (12 + 8) = 2 × 20 = 40 m.
2. Tiling a Floor: A square room has a side of 4 m. How many square meters of tiles are needed?
   • Area = 4 × 4 = 16 m².

Cost-Based Problems

• Example: If fencing costs ₹50 per meter, and a rectangular field is 10 m by 6 m,
  Perimeter = 2 × (10 + 6) = 32 m,
  Cost = 32 × 50 = ₹1600.

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7. Key Formulas Summary

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8. Solved Examples

1. Rectangle Problem: A rectangle has a length of 8 cm and a breadth of 5 cm. Find its perimeter and area.
   • Perimeter = 2 × (8 + 5) = 2 × 13 = 26 cm.
   • Area = 8 × 5 = 40 cm².
2. Triangle Problem: A triangle has a base of 10 cm and a height of 6 cm. Find its area.
   • Area = ½ × 10 × 6 = ½ × 60 = 30 cm².
3. Circle Problem: A circle has a radius of 14 cm. Find its circumference and area (use π = 22/7).
   • Circumference = 2 × 22/7 × 14 = 88 cm.
   • Area = 22/7 × 14 × 14 = 22/7 × 196 = 616 cm².

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9. Practice Questions

1. Find the perimeter of a square with a side of 9 cm.
2. A rectangle has a length of 15 m and a breadth of 10 m. Calculate its area and perimeter.
3. A triangle has sides 7 cm, 8 cm, and 9 cm. What is its perimeter?
4. Find the area of a parallelogram with a base of 12 cm and height of 5 cm.
5. A circle has a radius of 21 cm. Find its circumference and area (use π = 22/7).

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10. Tips for Students

• Always write the formula first before substituting values.
• Double-check units (convert if necessary).
• For circles, choose π = 22/7 when the radius is divisible by 7 for simpler calculations.
• Draw diagrams for word problems to visualize the shape.

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