Heron’s formula is one of 5 triangles formulas in the geometry section of this library, and it is used at high school level.
Why heron’s formula works
It is the usual half-base-times-height rule after the three sides have been used to work the height out. The semiperimeter appears because each of the three subtractions measures how much slack a side has against the other two, and multiplying those slacks together is what survives the algebra.
What each symbol means
$a,b,c$ are side lengths and $s$ is semiperimeter.
Heron’s formula: when it holds
Positive side lengths must satisfy the triangle inequalities.
When it stops applying
The three lengths must actually be able to close into a triangle. With sides 2, 3 and 9 the longest side exceeds the other two combined, one factor turns negative, and the formula asks for the square root of −280.
Heron’s formula: a worked example
Sides $3,4,5$ give $s=6$ and $A=\sqrt{6\cdot3\cdot2\cdot1}=6$.
The mistake to avoid
What people do: Putting the full perimeter into the formula in place of half of it.
Why it goes wrong: For sides 3, 4 and 5 that produces a root of 6048, about 78, against a true area of 6.
Do this instead: Halve the perimeter first, then subtract each of the three sides from that halved value.
Heron’s formula: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a,b,c$ are side lengths and $s$ is semiperimeter.
- Check the conditions before substituting. Positive side lengths must satisfy the triangle inequalities.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most geometry slips.
Where this formula fits
- Subject
- Geometry formulas — 30 entries in this library
- Topic
- Triangles
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where heron’s formula comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Area of a Triangle — the lesson behind this formula: choose a base-height, trigonometric, or side-length formula.
- Geometry Calculator — check your substitution and the value it produces.
- Study geometry — the subject guide that explains the ideas these formulas compress.
- Geometry Practice — questions that make you retrieve the formula instead of recognising it.
- All 30 geometry formulas — the full grouped reference, or the complete formula library.
Questions about heron’s formula
What does the semiperimeter mean?
It is half the sum of the three sides. It appears once by itself and once inside each of the three subtractions under the square root.
Is the result exact or an estimate?
It is exact. The square root may be an irrational number, but the value it names is the true area rather than an approximation of it.
Does it agree with the base-and-height rule for a right triangle?
Yes. For sides 3, 4 and 5 it returns 6, the same answer as half of 3 times 4, which is a good way to check your arithmetic.
Why is the answer so rarely a whole number?
The product of the four factors is only occasionally a perfect square. Triangles where it works out neatly are called Heronian triangles.