What it does
If you know the base and height of a triangle, its area is ½ × base × height. But often you only know the three sides, and finding the height is awkward. Heron’s formula gives the area straight from the sides a, b and c:
- first the semi-perimeter s = (a + b + c)/2,
- then area = √[s(s − a)(s − b)(s − c)].
This page draws the triangle for any three sides you choose (or tells you, with a picture, when the sides cannot meet), works out the formula step by step, names the kind of triangle (equilateral, isosceles or scalene; acute, right or obtuse), and checks the answer against ½ × base × height using the height it draws.
It follows the Class 9 chapter Heron’s formula, which many state boards still teach.
For right triangles there is the quicker ½ × leg × leg, which you can explore with pythagoras proof playground. Triangles with the same shape but different sizes are in similar triangles, and the height on a side is a sine away in trig ratio explorer.
How to use it
- Move the sliders for a, b and c, or type exact values. Side c is drawn as the base, from A to B; a is BC and b is CA.
- Or drag the top corner C. The base stays fixed and a and b change.
- Read the steps: s, the three brackets, their product and its square root.
- Under the steps, the same area comes out of ½ × base × height.
- If a red picture says the sides can’t make a triangle, one side is as long as (or longer than) the other two put together.
A worked example
The page starts with a = 13, b = 15 and c = 14.
- s = (13 + 15 + 14)/2 = 21.
- s − a = 8, s − b = 6, s − c = 7.
- s(s − a)(s − b)(s − c) = 21 × 8 × 6 × 7 = 7 056.
- Area = √7 056 = 84 square units.
- Check: the height on the base 14 is 2 × 84 / 14 = 12, and ½ × 14 × 12 = 84.
Try 10, 10, 10: s = 15 and the area is √(15 × 5 × 5 × 5) ≈ 43.30, the same as (√3/4) × 10².
Tips
- Always check the triangle rule first: each side must be shorter than the sum of the other two.
- With whole-number sides, the product under the square root is often a perfect square in textbook questions. Factorise it to take the root without a calculator.
Check yourself
Four fresh questions each time: an area from three sides, an equilateral triangle, sides in a ratio with a given perimeter, and whether three lengths make a triangle. Type the answer, press Check, and open the worked answer if you need it. Nothing is saved.
FAQ
Who was Heron? Heron of Alexandria, a Greek mathematician and engineer of the first century, wrote the formula down in his book Metrica.
Is this from the NCERT book? It covers the same topic with fibub’s own examples. fibub is not part of NCERT.
Where the numbers come from
- Heron's formula, the triangle inequality and ½ × base × height are standard geometry. The numerically stable ordering of the formula follows W. Kahan's well-known note on triangle areas. Written for fibub.
- Checked against Python for ten triangles with whole-number areas (13-14-15 → 84, 5-12-13 → 30, ...) and against ½ × base × height for 300 random triangles.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.