When should I use Heron's formula instead of base times height?

Use Heron's formula when you know all three side lengths but do not know the height. The base-times-height formula (A = 0.5 * base * height) requires knowing an altitude, which is often not directly given. Heron's formula only needs the three sides.

Heron's Formula Calculator

Calculate the area of a triangle from its three sides with Heron's formula. Get the semi-perimeter, area, perimeter, inradius, circumradius, all three altitudes, interior angles and triangle type, with an interactive diagram.

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Area from the sides: A = √(s(s−a)(s−b)(s−c)), s = half perimeter

Enter the three side lengths below to get the area, semi-perimeter, heights and angles, with the steps.

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About Heron's Formula Calculator

The Heron's Formula Calculator computes the area of any triangle when you know all three side lengths. Enter sides $a$, $b$, and $c$, and instantly get the area using Heron's formula $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$ is the semi-perimeter. The calculator also provides the perimeter, all three altitudes, interior angles, inradius, circumradius, and triangle classification with step-by-step formulas and an interactive diagram.

What Is Heron's Formula?

Heron's formula (sometimes called Hero's formula) is named after Hero of Alexandria, a Greek mathematician who lived in the 1st century AD. It allows you to calculate the area of a triangle using only the three side lengths — no angles or heights needed. The formula is:

$$A = \sqrt{s(s-a)(s-b)(s-c)}$$

where $s = \frac{a+b+c}{2}$ is the semi-perimeter (half the perimeter). This elegant formula works for all types of triangles — equilateral, isosceles, scalene, acute, right, and obtuse.

Real-World Applications

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Construction

Calculate roofing, flooring, and lot areas from measured sides

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Surveying

Find land area from boundary measurements

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Design

Compute triangular panel areas for fabrication

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Education

Verify geometry homework and exam solutions

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Navigation

Triangulation area in GPS coordinate mapping

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Game Dev

Triangle mesh area calculations in 3D modeling

Key Formulas

Property	Formula	Description
Semi-perimeter	$s = \frac{a+b+c}{2}$	Half the triangle's perimeter
Area (Heron's)	$A = \sqrt{s(s-a)(s-b)(s-c)}$	Area from three side lengths
Altitude	$h_a = \frac{2A}{a}$	Height perpendicular to side $a$
Inradius	$r = \frac{A}{s}$	Radius of inscribed circle
Circumradius	$R = \frac{abc}{4A}$	Radius of circumscribed circle
Angle (Law of Cosines)	$\angle A = \arccos\left(\frac{b^2+c^2-a^2}{2bc}\right)$	Interior angle opposite side $a$

How to Use the Heron's Formula Calculator

Enter side lengths: Type the three side lengths (a, b, c) of your triangle. You can use decimal values or click a quick example button to auto-fill sample values.
Preview the triangle: As you type, the live triangle preview updates in real-time, showing the actual shape and proportions along with a quick area estimate.
Click Calculate Area: Press the button to compute all results. The calculator validates the triangle inequality automatically.
Review results: See the area, perimeter, semi-perimeter, all three altitudes, interior angles, inradius, circumradius, and triangle classification. Use the diagram toggles to show or hide heights, angles, the incircle, and circumcircle.

Triangle Inequality Theorem

Not every combination of three positive numbers can form a triangle. The triangle inequality theorem requires that the sum of any two sides must be greater than the third side: $a + b > c$, $a + c > b$, and $b + c > a$. If any of these conditions is not met, the three lengths cannot form a valid triangle. This calculator automatically checks this condition and displays an error message if the sides are invalid.

Triangle Types

Triangles can be classified by their sides and angles. By sides: an equilateral triangle has all three sides equal, an isosceles triangle has exactly two sides equal, and a scalene triangle has all three sides different. By angles: an acute triangle has all angles less than 90°, a right triangle has one angle exactly 90°, and an obtuse triangle has one angle greater than 90°. This calculator determines both classifications automatically.

Inradius and Circumradius

The inradius ($r$) is the radius of the inscribed circle — the largest circle that fits inside the triangle, tangent to all three sides. It is calculated as $r = A/s$. The circumradius ($R$) is the radius of the circumscribed circle — the circle passing through all three vertices. It is calculated as $R = abc/(4A)$. These two radii are related by Euler's formula: the distance between the incenter and circumcenter is $\sqrt{R^2 - 2Rr}$.

FAQ

Can the calculator give the exact area in square-root form?

Yes. When the sides are whole numbers or simple decimals, the area is also shown in exact simplified radical form next to the decimal value, for example 6√6 for sides 5, 6 and 7, or 25√3/4 for an equilateral triangle with side 5.

How do I find the area of an irregular plot with Heron's formula?

Split the plot into triangles along diagonals you can measure, then enter every triangle in the Several Triangles box, one per line as a, b, c. The calculator adds up the areas and converts the total to square feet, square meters, acres and hectares.

What is Heron's formula?

Heron's formula calculates the area of a triangle when you know all three side lengths. The formula is A = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter. It works for any triangle — acute, obtuse, or right — without needing to know an angle or height.

When should I use Heron's formula instead of base × height?

Use Heron's formula when you know all three side lengths but do not know the height. The base-times-height formula (A = ½ × base × height) requires knowing an altitude, which is often not directly given in a problem. Heron's formula only needs the three sides.

What is the triangle inequality theorem?

The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side. If a+b ≤ c, a+c ≤ b, or b+c ≤ a, the three lengths cannot form a valid triangle. This calculator checks this condition automatically and alerts you if the sides are invalid.

What are the inradius and circumradius of a triangle?

The inradius (r) is the radius of the largest circle that fits inside the triangle (inscribed circle), calculated as r = A/s where A is area and s is semi-perimeter. The circumradius (R) is the radius of the circle passing through all three vertices (circumscribed circle), calculated as R = abc/(4A).

Can Heron's formula work for all types of triangles?

Yes. Heron's formula works for equilateral, isosceles, and scalene triangles, as well as acute, right, and obtuse triangles. It is a universal formula for any valid triangle as long as you know the three side lengths.

What is the area of a triangle with sides 13, 14, and 15?

The area is 84 square units. The semi-perimeter is (13 + 14 + 15) ÷ 2 = 21, so Heron’s formula gives √[21 × (21−13) × (21−14) × (21−15)] = √7056 = 84. The result uses the square of whatever unit you used for the sides, such as square feet if all three measurements are in feet.

How do I find a triangle’s height when I only know its three sides?

First calculate the area from the three sides, then divide twice the area by the side you want to use as the base: height = 2A ÷ base. For a triangle with sides 13, 14, and 15, the area is 84, so the altitude to the 14-unit side is 168 ÷ 14 = 12 units. The calculator reports all three altitudes, one for each side.

Why does Heron’s formula fail for my measured side lengths?

The three lengths must form a possible triangle: the sum of any two sides must be greater than the third. For instance, sides 2, 3, and 6 cannot make a triangle because 2 + 3 is less than 6. Measurements rounded from a real object can also fail this check if their rounding error is large. Recheck the sides and use consistent units before calculating.

Reference this content, page, or tool as:

"Heron's Formula Calculator" at https://MiniWebtool.com/heron-s-formula-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-04

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

Heron's Formula Calculator

About Heron's Formula Calculator

What Is Heron's Formula?

Real-World Applications

Key Formulas

How to Use the Heron's Formula Calculator

Triangle Inequality Theorem

Triangle Types

Inradius and Circumradius

FAQ

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Property	Formula	Description
Semi-perimeter	\(s = \frac{a+b+c}{2}\)	Half the triangle's perimeter
Area (Heron's)	\(A = \sqrt{s(s-a)(s-b)(s-c)}\)	Area from three side lengths
Altitude	\(h_a = \frac{2A}{a}\)	Height perpendicular to side \(a\)
Inradius	\(r = \frac{A}{s}\)	Radius of inscribed circle
Circumradius	\(R = \frac{abc}{4A}\)	Radius of circumscribed circle
Angle (Law of Cosines)	\(\angle A = \arccos\left(\frac{b^2+c^2-a^2}{2bc}\right)\)	Interior angle opposite side \(a\)