CBSE Class 9 Mathematics Heron's Formula Notes Set 14

Perimeter and Area of a Triangle

The perimeter is the sum of the three sides. The semi-perimeter is s = (a + b + c)/2. The area is measured in square units, and the units of length must be the same before calculating.

Heron's Formula

When the three sides a, b, c of a triangle are known, the area is \( \sqrt{s(s - a)(s - b)(s - c)} \). It is useful when the height is not given.

Special Triangles

Right triangle: area = ½ × product of the legs and hypotenuse = √(sum of squares of legs). Equilateral triangle of side a: height = (√3/2)a and area = (√3/4)a². Isosceles triangle with equal sides a and base b: height = √(a² − b²/4) and area = (b/4)√(4a² − b²).

Useful Observations

Doubling all sides makes the area four times. The altitude on a side is 2 × area ÷ that side, so the longest altitude falls on the shortest side. In an equilateral triangle the perpendiculars from an interior point add up to the height.

Solved Example

Find the area of a triangle with sides 13 cm, 14 cm and 15 cm.
s = 21 cm. Area = √(21 × 8 × 7 × 6) = √7056 = 84 cm².

Solved Example 2

The sides containing the right angle of a right triangle differ by 14 cm and its area is 120 cm². Find the perimeter.
x(x − 14)/2 = 120 gives x = 24, so the legs are 24 cm and 10 cm and the hypotenuse is 26 cm. Perimeter = 60 cm.