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Heron's Formula

Heron's formula gives the area of a triangle when the lengths of all three sides are known.

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Student-friendly explanation

When perpendicular height is not given, first find the semi-perimeter, then use the product involving the three side differences. It is especially useful for scalene triangles.

How to write this in exams

  1. 1

    Start with the exact idea

    Heron's formula gives the area of a triangle when the lengths of all three sides are known.

  2. 2

    Then show how to use it

    Add the three sides, divide by 2 to get s, calculate s-a, s-b, s-c, substitute, simplify the square root, and write square units.

  3. 3

    Add one concrete example

    For sides 13 cm, 14 cm, and 15 cm, s = 21 cm and area = √(21 × 8 × 7 × 6) = 84 cm².

  4. 4

    Avoid this incomplete answer

    Using 30 instead of 15 as s for sides 5, 12, 13 gives a wrong product because s must be half the perimeter.

Definition

Heron's formula gives the area of a triangle when the lengths of all three sides are known.

Example

For sides 13 cm, 14 cm, and 15 cm, s = 21 cm and area = √(21 × 8 × 7 × 6) = 84 cm².

Rule to remember

s = (a + b + c)/2; Area = √[s(s - a)(s - b)(s - c)]. Use when all three sides are known.

Memory hook

Heron's formula starts with half the boundary.

Examples and method

Worked example

For sides 5 cm, 12 cm, 13 cm, s = 15. Area = √(15 × 10 × 3 × 2) = √900 = 30 cm².

Method to apply

Add the three sides, divide by 2 to get s, calculate s-a, s-b, s-c, substitute, simplify the square root, and write square units.

Diagram support

A triangle labelled a, b, c is enough; no height is required for Heron's formula.

How CBSE asks it

CBSE-style questions often give three sides and ask area, or give a field shaped like a triangle.

Avoid common mistakes

Common confusion

Students sometimes use the full perimeter in place of semi-perimeter s.

Common wrong answer

Using 30 instead of 15 as s for sides 5, 12, 13 gives a wrong product because s must be half the perimeter.

Exam tip

Write s = (a + b + c)/2 clearly before applying the square-root formula.

Quick check

For a triangle with sides 6 cm, 8 cm, and 10 cm, what is the semi-perimeter?

The semi-perimeter is (6 + 8 + 10)/2 = 12 cm, which is the value of s used in Heron's formula.

Study the heron's formula diagram carefully

Use the labelled diagram to keep heron's formula clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of heron's formula in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on heron's formula.

Revision cue

Revise heron's formula through the labels before writing the answer.

Answer writing and exam use

1-mark answer

Heron's formula gives the area of a triangle when the lengths of all three sides are known.

2-mark answer

Heron's formula gives the area of a triangle when the lengths of all three sides are known. s = (a + b + c)/2; Area = √[s(s - a)(s - b)(s - c)]. Use when all three sides are known. For sides 13 cm, 14 cm, and 15 cm, s = 21 cm and area = √(21 × 8 × 7 × 6) = 84 cm².

3-mark answer

When perpendicular height is not given, first find the semi-perimeter, then use the product involving the three side differences. It is especially useful for scalene triangles. s = (a + b + c)/2; Area = √[s(s - a)(s - b)(s - c)]. Use when all three sides are known. For sides 5 cm, 12 cm, 13 cm, s = 15. Area = √(15 × 10 × 3 × 2) = √900 = 30 cm². CBSE-style questions often give three sides and ask area, or give a field shaped like a triangle. Using 30 instead of 15 as s for sides 5, 12, 13 gives a wrong product because s must be half the perimeter.
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