Key takeaways
- Heron's formula is A = √(s(s-a)(s-b)(s-c)), where s = (a + b + c) / 2.
- It finds a triangle's area from three side lengths alone, no height needed.
- s is the semi-perimeter, half the total perimeter.
- The three sides must actually form a valid triangle, or the formula breaks down.
In this guide
What is Heron's formula?
Most triangle area formulas need a base and a height. Heron's formula instead works with just the three side lengths, making it useful for land plots or triangles where the height is awkward to measure.
Why does Heron's formula matter?
It's the go-to tool whenever you have three measured distances but no clean way to find a perpendicular height — very common in land surveying and construction.
As a sanity check, Heron's formula agrees perfectly with the simpler½bhformula: a 3-4-5 right triangle givess = 6, soA = √(6 × 3 × 2 × 1) = √36 = 6, exactly matching½ × 3 × 4 = 6calculated the usual way.
The formula
| Symbol | What it means | Example |
|---|---|---|
| A | The area you want to find | 26.83 m² |
| a, b, c | The triangle's three side lengths | 7 m, 8 m, 9 m |
| s | The semi-perimeter: half the perimeter | 12 m |
How to find the area step by step
- 1Measure all three side lengths. a, b, and c.
- 2Add the three sides and divide by 2. This gives the semi-perimeter, s.
- 3Subtract each side from s, one at a time.
(s-a),(s-b),(s-c). - 4Multiply s by all three results.
- 5Take the square root of that product to get the area.
Now try it yourself. Change the three side lengths in the calculator below and watch the area update, with every step shown.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting to divide by 2 for the semi-perimeter. s is half the perimeter, not the full perimeter.
- Using this when you already have a base and height. The regular triangle area calculator (½bh) is simpler in that case.
- Skipping the final square root. The product
s(s-a)(s-b)(s-c)is the squared area, not the area itself.
Tips and tricks
- If your triangle happens to be right-angled, cross-check your Heron's-formula answer against the simpler ½bh calculator — they should match.
- Once you know it's a right triangle, the Pythagorean theorem calculator can verify the sides fit together correctly.
Where you'll use it in real life
- Land surveying: measuring the area of an irregular triangular plot using only its three boundary lengths.
- Construction: checking a triangle's area without needing to find a perpendicular height.
- Navigation: calculating triangular regions from known distances between three points.
- Design: verifying triangular panel sizes when only the edge lengths are known.
Quick summary
To use Heron's formula, find the semi-perimeter by halving the sum of the three sides, then take the square root of s times each side subtracted from s. It works for any valid triangle, using only the three side lengths. Use the calculator above to check your own triangle instantly.
Worked example
A triangular plot of land has boundary sides measuring 7 meters, 8 meters, and 9 meters, with no easy way to measure a perpendicular height. What's the plot's area?
- 01s = (a + b + c) / 2
- 02s = (7 + 8 + 9) / 2 = 12
- 03A = √(s(s-a)(s-b)(s-c))
- 04A = √(12 × 5 × 4 × 3)
- 05A = √720
- 06A = 26.83 m²
These numbers are pre-filled in the calculator above.
Frequently asked questions
When should I use Heron's formula instead of ½bh?
Use Heron's formula when you know all three side lengths but don't have (or can't easily measure) the height. Use [½bh](/calculator/area-triangle) when you already have a base and a perpendicular height.
What happens if the three sides can't form a triangle?
If one side is longer than or equal to the sum of the other two, no triangle exists, and Heron's formula won't give a valid result — you'd be trying to take the square root of a negative number.
Does the order of the sides a, b, and c matter?
No. Heron's formula gives the same area no matter which side you label a, b, or c, since all three are treated the same way.
Can Heron's formula be used on any triangle, not just right triangles?
Yes, Heron's formula works on any triangle, right, acute, or obtuse, as long as the three side lengths are valid.
