Heron's Formula
Heron's formula lets you find the area of a triangle when you know only its three sides — no height needed. This Class 9 chapter introduces the semi-perimeter, the formula itself, and how to extend it to quadrilaterals by dividing them into triangles. It is short, formula-based and reliably scoring.
Learning objectives
- Compute the semi-perimeter of a triangle.
- Find the area of a triangle using Heron's formula.
- Apply the formula when the height is unknown.
- Find the area of a quadrilateral by splitting it into triangles.
- Use the formula in real-life area problems.
Key concepts
Semi-perimeter
For a triangle with sides a, b and c, the semi-perimeter is s = (a + b + c)/2. It is half the perimeter and is the starting point for Heron's formula.
Heron's formula
The area of the triangle is Area = √[s(s − a)(s − b)(s − c)]. This works for any triangle, scalene, isosceles or equilateral, using only its side lengths.
Why it is useful
When the height of a triangle is not given (as in many real figures and fields), the usual ½ × base × height cannot be applied directly, but Heron's formula gives the area straight from the sides.
Area of quadrilaterals
A quadrilateral can be divided into two triangles by a diagonal. Find the area of each triangle (using Heron's formula if needed) and add them to get the total area.
Important formulas
Semi-perimeter
s = (a + b + c) ÷ 2
Heron's formula
Area = √[s(s − a)(s − b)(s − c)]
Equilateral triangle area
Area = (√3 ÷ 4) × side²
Key definitions
- Semi-perimeter
- Half the perimeter of a triangle, s = (a + b + c)/2.
- Heron's formula
- A formula giving a triangle's area from its three side lengths.
- Perimeter
- The total length of the boundary of a figure.
Solved examples
Q1. Find the area of a triangle with sides 3 cm, 4 cm and 5 cm.
Solution: s = (3 + 4 + 5)/2 = 6. Area = √[6(6 − 3)(6 − 4)(6 − 5)] = √[6 × 3 × 2 × 1] = √36 = 6 cm².
Q2. Find the area of a triangle with sides 13 cm, 14 cm and 15 cm.
Solution: s = (13 + 14 + 15)/2 = 21. Area = √[21(21 − 13)(21 − 14)(21 − 15)] = √[21 × 8 × 7 × 6] = √7056 = 84 cm².
Q3. Find the area of an equilateral triangle of side 6 cm.
Solution: Area = (√3/4) × 6² = (√3/4) × 36 = 9√3 cm² ≈ 15.59 cm².
Common mistakes to avoid
- Using the perimeter instead of the semi-perimeter in the formula.
- Forgetting to take the square root at the end.
- Subtracting the wrong side from s.
- Mixing up units (area is in square units).
Heron's Formula — MCQ Quiz
12 questions with instant feedback. Use number keys 1–4 to answer.
The semi-perimeter of a triangle with sides a, b, c is:
Practice questions
Short answer
Write Heron's formula.
Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2.
Find s for a triangle with sides 7, 8, 9.
s = (7 + 8 + 9)/2 = 12.
What is the area formula for an equilateral triangle of side a?
(√3/4)a².
Long answer
Find the area of a triangular park with sides 40 m, 24 m and 32 m.
s = (40 + 24 + 32)/2 = 48. Area = √[48(48 − 40)(48 − 24)(48 − 32)] = √[48 × 8 × 24 × 16] = √147456 = 384 m².
Find the area of a quadrilateral ABCD in which a diagonal divides it into two triangles of sides (3, 4, 5) and (5, 12, 13).
Triangle 1: s = 6, area = √[6·3·2·1] = 6. Triangle 2: s = 15, area = √[15·10·3·2] = 30. Total area = 6 + 30 = 36 square units.
HOTS (Higher Order Thinking)
The sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 36 cm. Find its area.
Sides are 9, 12 and 15 cm. s = 18; Area = √[18(18 − 9)(18 − 12)(18 − 15)] = √[18 × 9 × 6 × 3] = √2916 = 54 cm².
Why does Heron's formula always give the same answer as ½ × base × height for a triangle?
Both compute the same area; Heron's formula is derived from the standard area formula and the side lengths, so for any valid triangle the two agree.
Quick revision
Revision notes
- s = (a + b + c)/2.
- Area = √[s(s − a)(s − b)(s − c)].
- Use when the height is unknown.
- Quadrilateral area = sum of two triangle areas (split by a diagonal).
Key takeaways
- Always use the semi-perimeter, not the full perimeter.
- Heron's formula needs only the three sides.
- Split quadrilaterals into triangles to use it.
Frequently asked questions
When should I use Heron's formula?
When you know all three sides of a triangle but not its height.
What is the semi-perimeter?
Half the perimeter, s = (a + b + c)/2 — the value plugged into Heron's formula.
Can Heron's formula find the area of a quadrilateral?
Indirectly — divide the quadrilateral into two triangles with a diagonal, find each area, and add them.