Heron's Formula Notes
Find the area of a triangle from its three sides, check whether the triangle exists, and divide a quadrilateral into triangles. Includes worked examples, common mistakes and practice with full solutions. Match these topic notes to the textbook edition used by your school.
- 3 min read
- 6 practice questions
- Aligned to CBSE 2025–26 syllabus
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- Updated Sept 2026
Use this map for a first read, then return to the checks and formulas during revision.
- 1.Before you begin: area and valid triangles
- 2.Semiperimeter and Heron's formula
- 3.Worked example: sides 13 cm, 14 cm and 15 cm
- 4.Worked example: a quadrilateral split by a diagonal
- 5.Common mistakes and a final check
- 2 key formulas
- 2 concept checks
Before you begin: area and valid triangles
Area measures the region inside a shape and is expressed in square units. The familiar triangle formula, half × base × perpendicular height, is convenient when a height is known. Heron's formula is useful when all three side lengths are known instead.
Check the triangle inequality before using the formula: the sum of any two sides must exceed the third. For positive sides sorted from shortest to longest, it is enough to check that the two shorter sides add to more than the longest. Lengths 2, 3 and 5 form a straight-line boundary, not a triangle with positive area.
Semiperimeter and Heron's formula
Let the three side lengths be a, b and c. First calculate the semiperimeter s, which is half the perimeter. Then multiply s, s − a, s − b and s − c before taking the square root.
The product inside the root has units of length to the fourth power, so its square root has square units, as an area should. Keep exact values until the last step to avoid unnecessary rounding.
Worked example: sides 13 cm, 14 cm and 15 cm
The perimeter is 42 cm, so s = 21 cm. The three differences are 8 cm, 7 cm and 6 cm. The area is √(21 × 8 × 7 × 6) = √7056 = 84 cm².
Check the result using base 14 cm: the corresponding height must be 2 × 84 ÷ 14 = 12 cm. This is a perpendicular height, not one of the given sloping sides.
Worked example: a quadrilateral split by a diagonal
Suppose diagonal AC = 13 m divides a quadrilateral into two triangles with sides AB = 5 m, BC = 12 m and AC = 13 m, and AD = 13 m, CD = 10 m and AC = 13 m. Assume B and D lie on opposite sides of AC so their interiors do not overlap.
For triangle ABC, s = 15 and the area is √(15 × 10 × 3 × 2) = 30 m². For triangle ACD, s = 18 and the area is √(18 × 5 × 8 × 5) = 60 m². The quadrilateral has area 30 + 60 = 90 m².
Four side lengths alone generally do not fix a quadrilateral's area. You need enough information, such as a diagonal, to determine the component triangles.
Common mistakes and a final check
Do not confuse perimeter with semiperimeter. Do not multiply half the base by a sloping side unless that side is perpendicular to the base. Convert units before substituting, and remember to take the square root of the whole product.
If the result under the root is negative for supposed side lengths, check the triangle inequality and your arithmetic. If you double every side, the area becomes four times as large, not twice as large. Use this scaling check to spot implausible answers.
Heron's Formula: revision sheet
Close the notes and explain each idea in your own words. Then check the formulas and attempt the examples before opening their solutions.
Recall checklist
- Before you begin: area and valid triangles
- Semiperimeter and Heron's formula
- Worked example: sides 13 cm, 14 cm and 15 cm
- Worked example: a quadrilateral split by a diagonal
- Common mistakes and a final check
For each topic: state the rule, explain when it applies, give an example and identify one mistake to avoid. Return to any topic you cannot explain without looking.
Formula recap (2)
Semiperimeter and Heron's formula
Semiperimeter and Heron's formula
Explain your method
Write the given information, choose a method, show the steps, then check units and whether the answer makes sense.
Check worked solution 1
Check worked solution 2
Turn mistakes into revision
Record the question, the step that went wrong and the corrected reason. Try it again tomorrow without the solution, then revisit it after a week. A correct guess still needs an explanation.
Practice and Revision
Test your understanding with quick chapter-level practice.
Chapter Q&A
Does Heron's formula work only for right triangles?
No. It works for any triangle with valid positive side lengths, including acute, right and obtuse triangles.
Why do I get zero for sides 2, 3 and 5?
The lengths satisfy equality instead of the strict triangle inequality. They form a degenerate straight-line figure with zero area, not an ordinary triangle.
Can I use Heron's formula when only a base and height are known?
Use half × base × perpendicular height directly. Heron's formula requires all three side lengths, so it adds unnecessary work in this case.