We know the area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). But what if
the height is not given and only the lengths of the three sides are known?
This is where Heron's Formula (given by Hero of Alexandria) saves the day!
Formula Type
When to use?
Expression
General Formula
When Base & Height are known
\( \frac{1}{2} \times b \times h \)
Heron's Formula
When 3 Sides (\(a, b, c\)) are known
\( \sqrt{s(s-a)(s-b)(s-c)} \)
Key Definitions (NCERT Language)
Semi-perimeter (s): Half of the perimeter of a triangle.
Heron’s Formula: A formula to find area of a triangle when all three sides are
known.
Hero of Alexandria: Greek mathematician who discovered this formula.
Triangle Validity Rule ⚠️
Three sides can form a triangle ONLY IF:
\( a + b > c \)
\( b + c > a \)
\( c + a > b \)
❗ CBSE often asks: “Check whether triangle is possible or not”
2. The Formula Explained
To find the area using three sides \( a, b, \) and \( c \):
Step 1: Find Semi-perimeter (\( s \)):
\( s = \frac{a + b + c}{2} \)
Step 2: Apply Formula:
\( Area = \sqrt{s(s-a)(s-b)(s-c)} \)
Special Cases Using Heron’s Formula
Triangle Type
Side Condition
Area Formula
Equilateral
a = b = c
\( \frac{\sqrt{3}}{4} a^2 \)
Isosceles
Two sides equal
Use Heron’s Formula
Scalene
All sides different
Use Heron’s Formula
Heron's Laboratory
Enter side lengths to calculate area step-by-step.
1. Check Validity:...
2. Semi-perimeter (s):...
3. Differences (s-a, s-b, s-c):...
4. Final Area:...
Solved Example (Board Pattern)
Question: Find the area of a triangle whose sides are 5 cm, 6 cm and 7 cm.
Show Solution
Step 1: Find semi-perimeter
\( s = \frac{5+6+7}{2} = 9 \)
Step 2: Apply Heron’s formula
\( \text{Area} = \sqrt{9(9-5)(9-6)(9-7)} \)
\( = \sqrt{9 \times 4 \times 3 \times 2} \)
\( = \sqrt{216} = 14.7 \, \text{cm}^2 \)
3. Application: Quadrilaterals
Heron's formula can be used to find the area of a quadrilateral by dividing it into two triangles.
Visual Splitter
Click "Split" to divide the quadrilateral.
Word Problems Using Heron’s Formula
Find area of triangular park
Find cost of leveling triangular field
Find area of land given side lengths
💡 Tip: Always convert units first (m → cm if needed)
Common Mistakes to Avoid ❌
❌ Forgetting to divide perimeter by 2
❌ Applying formula without checking triangle validity
❌ Wrong unit for area (always square units)
❌ Rounding too early in calculation
Assertion–Reason Practice
Assertion (A): Heron’s formula can be used when all sides of a triangle are known.
Reason (R): Semi-perimeter is calculated before finding area.
✔ Both A and R are true, and R explains A
One-Page Revision Sheet
\( s = \frac{a+b+c}{2} \)
\( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \)
Check triangle validity first
Used when height is NOT given
Can You Answer These Without Looking?
What is semi-perimeter?
When is Heron’s formula used?
What is area formula for equilateral triangle?
Why validity check is important?
Concept Mastery Quiz
1. The semi-perimeter 's' is given by:
2. For an equilateral triangle with side 'a', Area is: