Heron's Formula: Formula, Examples and Aptitude Problems

Heron's Formula finds the area of a triangle when the lengths of all three sides are known. It uses the semiperimeter and the three differences between the semiperimeter and the side lengths. This page covers the formula, triangle validity rule, calculation method, shortcuts and solved Heron's Formula problems.

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What is Heron's Formula?

Heron's Formula gives the area of a triangle when its three side lengths are known. If the sides are a, b and c, and the semiperimeter is s, then area = √[s(s − a)(s − b)(s − c)].

The semiperimeter is s = (a + b + c)/2. First calculate s, then substitute the three side lengths in the formula. The side lengths must satisfy the triangle inequality: a + b > c, b + c > a and c + a > b. For sides 3, 4 and 5, s = 6 and area = √[6 × 3 × 2 × 1] = 6 square units.

Heron's Formula Video Lessons

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Heron’s Formula for Triangle Area

Learn how to use Heron’s formula to calculate the area of a triangle when its three side lengths are known.

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Heron's Formula Formula & Tricks

Important Formulas

Heron's Formula
Area (Δ) = √[s(s − a)(s − b)(s − c)]

Here, a, b and c are the side lengths and s is the semiperimeter.

Semiperimeter
s = (a + b + c)/2

Add the three sides and divide the sum by 2 before using Heron's Formula.

Equilateral Triangle Shortcut
Area = (√3/4)a²

For an equilateral triangle, all three sides have length a.

Quick Tricks

Calculate the four factors before taking the square root

Find s, s − a, s − b and s − c separately. This reduces sign and substitution errors.

Example: For sides 13, 14 and 15, s = 21, so the factors are 21, 8, 7 and 6. Area = √(21 × 8 × 7 × 6) = √7056 = 84 square units.
Use the 3-4-5 triangle as a quick check

For sides 3, 4 and 5, the semiperimeter is 6, and Heron's Formula gives area 6 square units. This agrees with the right-triangle calculation (1/2 × 3 × 4).

Example: √[6(6 − 3)(6 − 4)(6 − 5)] = √(6 × 3 × 2 × 1) = 6.

Heron's Formula Concepts

Finding the Area from Three Sides

To find the area, calculate the semiperimeter first and then substitute it with the three sides in Heron's Formula.

For sides a, b and c, use s = (a + b + c)/2. Then calculate s − a, s − b and s − c. Multiply all four factors and take the positive square root because area cannot be negative.

Example: For sides 5, 6 and 7, s = 9. Therefore, area = √[9 × 4 × 3 × 2] = √216 = 6√6 square units.

Triangle Validity and Heron's Formula

Three lengths form a non-degenerate triangle only when the sum of any two sides is greater than the third side.

The conditions are a + b > c, b + c > a and c + a > b. If the largest side equals the sum of the other two, the figure is degenerate and its area is zero. If it is greater, no triangle can be formed.

Example: Sides 2, 3 and 5 do not form a proper triangle because 2 + 3 = 5. Sides 2, 3 and 6 are impossible because 2 + 3 < 6.

Solved Example with Integer Area

For sides 13, 14 and 15, the area is 84 square units.

The semiperimeter is s = (13 + 14 + 15)/2 = 21. Applying the formula gives area = √[21(21 − 13)(21 − 14)(21 − 15)] = √(21 × 8 × 7 × 6) = √7056 = 84 square units.

Example: The corresponding altitude to the side 14 is 2 × 84/14 = 12 units, since area = 1/2 × base × height.

Special Case: Equilateral Triangle

For an equilateral triangle of side a, the area is (√3/4)a².

Heron's Formula also produces this result because s = 3a/2 and each difference s − a equals a/2. Thus, area = √[(3a/2)(a/2)(a/2)(a/2)] = (√3/4)a².

Example: If each side is 8 cm, area = (√3/4) × 8² = 16√3 cm².

Heron's Formula Visual Guide

Use the diagram or chart below to understand the concept clearly.

Triangle with Three Known Sides

Visual guide: Triangle with Three Known Sides
a: 13 units b: 14 units c: 15 units
Heron's Formula uses all three side lengths of the triangle.

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Practice Heron's Formula Questions

Practise published questions related to this topic.

1The LCM of 53 × 82 × 14, 52 × 142 × 17, and 83 × 142 × 172 is:→ 2Ajay observes a 10% increase in sales in the first year and a 25% increase in the second year. If the current sales are ₹228800, what were the sales (in ₹) two years ago?→ 3What is the minimum number of times one needs to measure to get 298 litres of water from a tank, if the measuring cylinders have capacities 1 litre, 6 litres, 25 litres and 100 litres?→ 4The value of (√5+√3)/(√5-√3)+(√5-√3)/(√5+√3) is:→ 5Simplify: ((10/18))/((10/9)) ÷ (4/9 × 18/12 + 5/6) + 9/9 ÷ 27/15 of 15/9→ 6An article was sold for ₹ 1350 at a profit of 12.5%. What was its cost price?→ 7In a town with 75,000 voters, 95% cast their votes, all of which were valid. There were three candidates: A, B, and C. Candidate A received 40% of the valid votes, while candidate B received 12% fewer votes than A. What percentage of the valid votes did candidate C receive?→ 8A tank holds 27.5 liters of water. If a leak causes 3/8 of the water to drain out, how much water remains in the tank (in liters)?→ 9Which is the largest number that divides 38, 50, 74 and 98 to leave the same remainder in each case?→ 10The expenditure of Sudha is 175% more than her savings. If her expenditure decreases by 12% and savings increase by 40.5%, then by what percentage does her income increase?→

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Quick Revision Notes

Heron's Formula Revision Points

Use these formulas and checks for quick revision.

  • Semiperimeter: s = (a + b + c)/2.
  • Area: Δ = √[s(s − a)(s − b)(s − c)].
  • Use the positive square root because area is non-negative.
  • Check that the sum of any two sides is greater than the third side.
  • If a + b = c for the largest side c, the area is zero and the triangle is degenerate.
  • The area unit is the square of the side-length unit.

Related Topic Explainers

Area of Square → Perimeter of Square → Area of Rectangle → Perimeter of Rectangle → Area of Triangle → Perimeter of Triangle →

Heron's Formula FAQs

What is the formula for the area of a triangle using its three sides?

If the sides are a, b and c, and s = (a + b + c)/2, then area = √[s(s − a)(s − b)(s − c)].

What is the area of a triangle with sides 7, 8 and 9?

The semiperimeter is 12. Area = √[12 × 5 × 4 × 3] = √720 = 12√5 square units.

Can Heron's Formula be used for a right triangle?

Yes. For a 3-4-5 right triangle, it gives √(6 × 3 × 2 × 1) = 6 square units, which matches 1/2 × 3 × 4.

What happens if the sum of two sides equals the third side?

The triangle is degenerate, and its area is zero. For example, sides 3, 4 and 7 give s = 7 and a zero factor in Heron's Formula.

What are the units of area obtained from Heron's Formula?

The area is expressed in square units. If side lengths are in centimetres, the result is in cm²; if they are in metres, it is in m².

What is the area of an equilateral triangle using Heron's Formula?

For side length a, the area is (√3/4)a². For side 6 units, the area is 9√3 square units.

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