Class 11 Notes - Areas of Parallelograms and Triangles

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Areas of Parallelograms and Triangles

In this chapter, you will learn how to find the area of parallelograms and triangles, understand their properties, and apply these concepts to solve real-life and mathematical problems.

1. Parallelogram

  • A parallelogram is a quadrilateral with both pairs of opposite sides parallel and equal.
  • Examples: Rectangle, Rhombus, Square (special cases of parallelogram).

Area of a Parallelogram

  • Formula: Area = base × height
  • The height is the perpendicular distance from the base to the opposite side.
Example: Find the area of a parallelogram with base 8 cm and height 5 cm.
Solution: Area = 8 × 5 = 40 cm²

2. Triangle

  • A triangle is a polygon with three sides and three angles.
  • Types: Equilateral, Isosceles, Scalene, Right-angled, Acute, Obtuse.

Area of a Triangle

  • Formula: Area = (1/2) × base × height
  • The height is the perpendicular distance from the base to the opposite vertex.
Example: Find the area of a triangle with base 10 cm and height 6 cm.
Solution: Area = (1/2) × 10 × 6 = 30 cm²

Area Using Heron's Formula

  • For a triangle with sides a, b, c:
  • Semi-perimeter, s = (a + b + c) / 2
  • Area = √[s(s - a)(s - b)(s - c)]
Example: Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm.
Solution: s = (7 + 8 + 9)/2 = 12
Area = √[12 × 5 × 4 × 3] = √720 = ~26.83 cm²

3. Properties and Applications

  • Parallelograms on the same base and between the same parallels have equal areas.
  • Triangles on the same base and between the same parallels have equal areas.
  • Area concepts are used in land measurement, construction, and design.

4. Practice Problems

  1. Find the area of a parallelogram with base 12 cm and height 7 cm.
  2. A triangle has a base of 15 cm and a height of 10 cm. What is its area?
  3. Calculate the area of a triangle with sides 5 cm, 6 cm, and 7 cm using Heron's formula.
  4. Two parallelograms have the same base and height. Are their areas equal? Why?
  5. Give a real-life example where you need to calculate the area of a triangle or parallelogram.

5. Summary

  • Area of parallelogram = base × height
  • Area of triangle = (1/2) × base × height
  • Heron's formula is used when all sides of a triangle are known.
  • Understanding areas helps in solving practical problems in geometry and daily life.