If you are wondering how to calculate the area of a triangle knowing its three sides, you're at a right place - this Heron's formula calculator can do it in no time. Simply type in your given values - e.g. triangle sides - and our tool will show you the answer. If you are curious about Heron's area formula, you will read about it below. Also, the Heron's formula proofs may be found there, so keep scrolling to find out more about this useful, but not so well-known, formula.

Heron's area formula: equation for the area of triangle with 3 sides given

Heron's formula, also known as Hero's formula, is the formula to calculate triangle area given three triangle sides. It was first mentioned in Heron's book Metrica, written in ca. 60 AD, which was the collection of formulas for various objects surfaces and volumes calculation. The basic formulation is:

area = √(s * (s - a) * (s - b) * (s - c))

where s is the semiperimeter - half of triangle perimeter:

s = (a + b + c) / 2

However, other forms of this formula exist - if you don't want to calculate the semiperimeter by hand, you can use the formula with side lengths only:

area = 0.25 * √((a + b + c) * (-a + b + c) * (a - b + c) * (a + b - c))

Heron's formula proof

There are many ways to prove the Heron's area formula, but you need to know some geometry basics. You can use:

Other proofs also exist, but they are more complex or they use the laws which are not so popular (such as e.g. a trigonometric proof using the law of cotangents).

  • Algebraic proof

In this proof, we need to use the formula for the area of a triangle:

area = (c * h) / 2

All the values in the formula should be expressed in terms of the triangle sides: c is a side so it meets the condition, but we don't know much about our height. So to derive the Heron's formula proof we need to find the h in terms of the sides.

  1. From the Pythagorean theorem we know that:

    h² + (c - d)² = a² and h² + d² = b², according to the figure above

  2. Subtracting those two equations gives us:

    c² - 2 * c * d = a² - b² from which you can derive the formula for d in terms of the sides of the triangle:

    d = (-a² + b² + c²) / (2 * c)

  1. Next step is to find the height in terms of triangle sides. Use the Pythagorean theorem again:

    h² = b² - d²

    h² = b² - ((-a² + b² + c²) / (2 * c))² - it's already in terms of the sides, but let's try to reduce it to nicer form, applying the difference of squares identity:

    h² = ((2 * b * c) - a² + b² + c²) * ((2 * b * c) - a² + b² + c²) / (4 * c⁴)

    h² = ((b + c)² - a²) * (a² - (b - c)²) / (4 * c²)

    h² = (b + c - a) * (b + c + a) * (a + b - c) * (a - b + c) / (4 * c²)

  2. Apply this formula to first equation, the one for triangle area:

    area = (c * h) / 2 = 0.5 * c * h

    area = 0.5 * c * √((b + c - a) * (b + c + a) * (a + b - c) * (a - b + c) / (4 * c²))

    area = 0.25 * √((b + c - a) * (b + c + a) * (a + b - c) * (a - b + c)

Here you are! That's the Heron's area proof. Changing the final equation into the form using semiperimeter is a trivial task.

  • Trigonometric proof

Have a look at the picture - a, b, c are the sides of the triangle and α, β, γ are the angles opposing these sides. To find the proof of Heron's formula with trigonometry, we need to use another triangle area formula - given two sides and angle between them:

area = 0.5 * a * b * sin(γ)

  1. To derive the proof for Heron's formula in this case, we need to express sine of the angle in terms of the triangle sides. We can use the law of cosines:

    c² = a² + b² - 2 * a * b * cos(γ)

  2. Thanks to this law we get the side-dependent cosine. But how to find the formula for the sine? That's easy, just use the basic relation between the sine and cosine functions - Pythagorean trigonometric identity: sin²(γ) + cos²(γ) = 1, so the formula for cosine is

    sin(γ) = √(1 - cos²(γ)) = (√(4 * a² * b² - (a² + b² -c²)²)) / (2 * a * b)

  3. Now you can substitute the sine in triangle area formula:

    area = 0.5 * a * b * sin(γ)

    area = 0.25 * √(4 * a² * b² - (a² + b² - c²)²))

    area = 0.25 * √(2 * a * b - (a² + b² - c²) * (2 * a * b + (a² + b² - c²)))

    area = 0.25 * √(c² - (a - b)²) * (- c² + (a + b)²)

    area = 0.25 * √(c - a + b) * (c + a - b) * (- c + a + b) * (c + a + b)

which was to prove.

Great! Now you understand why this formula is valid, so don't wait any longer and give the Heron's formula calculator a try!

Hanna Pamuła, PhD student

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