Heron's Formula
About 7 min · Mathematical Reasoning · English
Time worked out from the lesson's 532 words at 150 words a minute, plus a minute for each question.
Heron's Formula is a powerful tool that lets you calculate the area of a triangle when you know all three side lengths but not the height. This is especially useful in IMO problems where triangles are embedded in complex figures or where finding the height directly is difficult. The formula extends naturally to quadrilaterals by splitting them into triangles along a diagonal.
Key ideas
- Semi-perimeter (s): Half the perimeter of the triangle. For a triangle with sides a, b, c: s = (a + b + c)/2. This value is central to Heron's Formula and must be calculated first.
- Heron's Formula: Area = √[s(s − a)(s − b)(s − c)], where s is the semi-perimeter and a, b, c are the three sides. Works for any triangle — scalene, isosceles, or equilateral.
- Units consistency: All three sides must be in the same unit (cm, m, etc.). The area will be in square units of that measurement.
- Quadrilateral decomposition: Any quadrilateral can be split into two triangles by drawing one diagonal. Find the area of each triangle separately using Heron's Formula, then add them.
- Triangle inequality check: Before applying the formula, verify that a + b > c, b + c > a, and c + a > b. If any inequality fails, those sides cannot form a triangle.
Watch out for
- Forgetting to halve the perimeter → Students write s = a + b + c instead of s = (a + b + c)/2. Always divide by 2 for the semi-perimeter.
- Sign errors in subtraction → Writing (s − a) as (a − s) or getting confused with negative values. Remember s is always greater than any individual side in a valid triangle, so (s − a), (s − b), (s − c) are all positive.
- Premature rounding → Taking √7056 ≈ 84.0 but rounding intermediate steps like 21 × 8 = 168.0. Keep exact values until the final answer to avoid compounding errors.
Quick reference
- Semi-perimeter first: s = (a + b + c)/2, then plug into Heron's Formula
- Area = √[s(s − a)(s − b)(s − c)] — memorize this exactly
- For quadrilaterals: split via diagonal, apply Heron to each triangle, sum the areas
- Always verify triangle inequality: sum of any two sides > third side
- Simplify under the square root completely before extracting the root
Check yourself: 3 questions
1. A triangular park has sides of length 13 m, 14 m and 15 m. Find the area of the park using Heron's formula.
2. A quadrilateral field ABCD has sides AB = 9 m, BC = 12 m, CD = 15 m and DA = 10 m. The diagonal AC divides it into two triangles. If triangle ABC has sides 9 m, 12 m and AC = 15 m, and triangle ACD has sides AC = 15 m, CD = 15 m and DA = 10 m, find the total area of the quadrilateral.
3. A triangle has sides in the ratio 5:12:13 and its perimeter is 60 cm. What is the area of the triangle?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
Read the full noteFlashcards for this topic
Saves this lesson and its 3 questions in this browser on this phone. No sign-in; nothing about you is saved or sent. A saved copy is not updated until you save it again. Parts that load from the server when you use them (the AI tutor, adaptive practice, sign-in) still need a connection.