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Heron's Formula Calculator

Heron's Formula Calculator is part of Toolspire and is ready as soon as this page loads. You need every side. Two sides and an angle are a different formula. The semi-perimeter is shown so the square root can be checked.

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On this page

What you can do here

Three sides, no angle

If you know the height instead, this is the wrong formula. Heron does not ask for a height.

Impossible triangles stop

The page returns nothing when one side is longer than the other two combined.

Semi-perimeter in the detail

You can recompute the product under the square root from s without guessing which step failed.

The sequence

How to use the Heron's Formula Calculator

  1. 1

    Use the same unit on every side

    Do not mix inches and centimetres. The area comes back in that unit squared.

  2. 2

    Try 3, 4 and 5

    The area should be 6. If it is not, the side boxes are not all filled.

  3. 3

    Stop if the page refuses

    A refusal means the lengths cannot meet. Measure again before you look for a rounding error.

Worth knowing

  • Heron's Formula Calculator on Toolspire: the semi-perimeter s = (a + b + c) / 2. Area = √[s(s − a)(s − b)(s − c)].
  • A 3-4-5 triangle has area 6. That is the right-triangle check, without using the right angle.
  • Sides that break the triangle inequality are rejected. 1, 2 and 10 do not form a triangle.
  • The unit of the area is the square of whatever unit the sides used. The page does not convert centimetres to metres.

Heron does not need the height

The usual school area is base times height over two. Heron replaces the height with a combination of all three sides, which is what you have when you measured a plot and did not drop a perpendicular. The square root is zero when the points are colinear, and it is not real when the triangle inequality fails. Those inputs are rejected rather than shown as zero area.

FAQ

Questions about Heron's Formula Calculator

Where does Heron's Formula Calculator run?

Heron's Formula Calculator runs in this tab on Toolspire. What you type or drop in is not sent to us and is not stored on a server.

What is Heron’s formula?

Area = √[s(s−a)(s−b)(s−c)], where s is half the perimeter. It needs all three sides and no angle.

What is the area of a 3-4-5 triangle?

6 square units. It is also a right triangle, and (3 × 4) / 2 gives the same 6.

Why did my sides get rejected?

One side is at least as long as the other two put together, so the ends cannot meet.

Can I use this for a right triangle?

Yes. You do not need to tell the page about the right angle. Pythagoras is available when you only know two sides of a right triangle.

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