MATHEMATICS • TRIANGLES

Heron’s Formula

Heron’s formula helps us find the area of a triangle when we know the lengths of all three sides—even when the height is not given.

It is especially useful for questions in which the three sides are known but no perpendicular height is provided.

What Is Heron’s Formula?

Usually, we find the area of a triangle using its base and perpendicular height. But sometimes, a question gives only the three side lengths. In that situation, we can use Heron’s formula.

\[\displaystyle A=\sqrt{s(s-a)(s-b)(s-c)}\]

\[\displaystyle s=\frac{a+b+c}{2}\]

Here:

  • a, b and c are the lengths of the three sides of the triangle.
  • s is the semiperimeter, which means half of the triangle’s perimeter.
  • Area is expressed in square units, such as cm² or m².

Understand the Formula with a Diagram

Triangle with side lengths a, b and c for Heron's formula A triangle has side a along the base, side b on the right, and side c on the left. The perpendicular height is shown as a dashed line to illustrate that Heron's formula does not require the height to be given. a c b height All three sides are used in Heron’s formula
The three side lengths are enough to calculate the area using Heron’s formula, provided they can form a triangle.

Steps to Use Heron’s Formula

  1. Write down the three side lengths: a, b and c.
  2. Find the perimeter by adding the three sides.
  3. Calculate the semiperimeter: \(s=\frac{a+b+c}{2}\).
  4. Substitute the values into \(A=\sqrt{s(s-a)(s-b)(s-c)}\).
  5. Simplify the expression and write the answer in square units.

Solved Example 1: A Triangle with Sides 3 cm, 4 cm and 5 cm

Question: Find the area of a triangle whose sides are 3 cm, 4 cm and 5 cm.

Step 1: Find the semiperimeter.

\[\displaystyle s=\frac{3+4+5}{2}=\frac{12}{2}=6\text{ cm}\]

Step 2: Apply Heron’s formula.

\[\displaystyle A=\sqrt{s(s-a)(s-b)(s-c)}\]

\[\displaystyle A=\sqrt{6(6-3)(6-4)(6-5)}\]

\[\displaystyle A=\sqrt{6\times3\times2\times1}=\sqrt{36}=6\text{ cm}^2\]

Answer: The area is 6 cm².

Solved Example 2: An Isosceles Triangle with Sides 15 cm, 15 cm and 10 cm

Question: Find the area of an isosceles triangle whose equal sides are 15 cm each and whose base is 10 cm.

Step 1: Find the semiperimeter.

\[\displaystyle s=\frac{15+15+10}{2}=\frac{40}{2}=20\text{ cm}\]

Step 2: Substitute in the formula.

\[\displaystyle A=\sqrt{20(20-15)(20-15)(20-10)}\]

\[\displaystyle A=\sqrt{20\times5\times5\times10}\]

\[\displaystyle A=\sqrt{5000}=50\sqrt{2}\text{ cm}^2\]

Approximately, \(A\approx70.71\text{ cm}^2\).

Answer: The area is 50√2 cm², or approximately 70.71 cm².

Solved Example 3: A Triangular Plot with Sides 13 m, 14 m and 15 m

Question: A triangular plot has side lengths 13 m, 14 m and 15 m. Find its area.

Step 1: Find the semiperimeter.

\[\displaystyle s=\frac{13+14+15}{2}=\frac{42}{2}=21\text{ m}\]

Step 2: Use Heron’s formula.

\[\displaystyle A=\sqrt{21(21-13)(21-14)(21-15)}\]

\[\displaystyle A=\sqrt{21\times8\times7\times6}\]

\[\displaystyle A=\sqrt{7056}=84\text{ m}^2\]

Answer: The area of the plot is 84 m².

Real-Life Uses of Heron’s Formula

1. Measuring triangular plots of land

If a plot is triangular and its three side lengths are known, Heron’s formula can help estimate its area without measuring the perpendicular height directly.

2. Surveying and field measurement

Surveyors may divide a larger area into triangles. When the sides of a triangle are measured, its area can be calculated and used as part of the total area estimate.

3. Construction and landscaping

Builders and landscapers may need the area of a triangular section for planning materials, garden beds, paving or layout work.

4. Design and craft work

The formula can help calculate the area of triangular pieces used in patterns, artwork, tiles and other designs when their side lengths are known.

For real projects, measurements should be taken carefully and the appropriate surveying or construction method should be followed.

Heron’s Formula or Base–Height Formula?

MethodFormulaWhen it is useful
Base–height method\(A=\frac12\times\text{base}\times\text{height}\)When the base and perpendicular height are known.
Heron’s formula\(A=\sqrt{s(s-a)(s-b)(s-c)}\)When all three side lengths are known.

Common Mistakes to Avoid

  • Forgetting to divide by 2: The semiperimeter is half the perimeter, not the full perimeter.
  • Using the wrong side values: Match each side carefully when substituting into the formula.
  • Leaving out brackets: Calculate each term such as (s − a) correctly before multiplying.
  • Forgetting square units: If the sides are measured in centimetres, the area is in cm².
  • Using impossible side lengths: The sum of any two sides of a triangle must be greater than the third side.

Practice Questions

  1. Find the area of a triangle with sides 5 cm, 12 cm and 13 cm.
  2. Find the area of a triangle with sides 7 cm, 8 cm and 9 cm.
  3. A triangular field has sides 20 m, 21 m and 29 m. Find its area.
Show answers

1. 30 cm²

2. 12√5 cm² (approximately 26.83 cm²)

3. 210 m²

Quick Revision

  • Heron’s formula: \(A=\sqrt{s(s-a)(s-b)(s-c)}\).
  • Semiperimeter: \(s=\frac{a+b+c}{2}\).
  • Use it when: The lengths of all three sides are known.
  • Units: Area is measured in square units.

Frequently Asked Questions

What is Heron’s formula?

Heron’s formula is a method for finding the area of a triangle when all three side lengths are known.

What does s mean in Heron’s formula?

The letter s represents the semiperimeter, which is half the sum of the three sides.

Can Heron’s formula be used for every triangle?

It can be used for any valid triangle when its three side lengths are known. The side lengths must satisfy the triangle inequality.

Why is Heron’s formula useful?

It helps calculate a triangle’s area without needing its perpendicular height.