How do you find the semiperimeter?

The semiperimeter s is half the perimeter: s = (a+b+c)/2. For a triangle with sides 5,6,7: s = (5+6+7)/2 = 9. The semiperimeter is used in Heron formula, inradius calculation, and many other triangle formulas.

What else can Heron's formula calculate?

Once area K is found, you can compute circumradius R = abc/(4K), inradius r = K/s, and each height h_a = 2K/a. Heron's formula is the gateway to many triangle properties from just the three side lengths.

What if the triangle inequality is not satisfied?

If any side is greater than or equal to the sum of the other two sides, a valid triangle does not exist. Heron formula would produce a negative value under the square root. The tool validates this before computing.

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Home›Geometry Tools›Heron's Formula Calculator

Heron's Formula Calculator

Enter three side lengths to compute triangle area using Heron's formula

Side a

Side b

Side c

Heron's Formula

s = (a + b + c) / 2

K = sqrt(s(s-a)(s-b)(s-c))

R = abc / (4K)

r = K / s

h_a = 2K / a

Heron's formula is one of the most famous formulas in geometry. It allows computing triangle area using only side lengths, without needing angles or height measurements.

⚠ Side lengths must satisfy the triangle inequality: a+b > c, a+c > b, b+c > a. Otherwise a valid triangle does not exist.

What Is Heron's Formula?

Heron's formula, attributed to Hero of Alexandria, computes triangle area from side lengths. It is especially useful when height is unknown or inconvenient to measure. The formula works for any triangle shape.

Semiperimeter

Half the perimeter: s = (a+b+c)/2. Averages the side lengths into a single measure used in the formula.

Square Root

The area is the square root of s(s-a)(s-b)(s-c). Each factor s-a, s-b, s-c is positive for valid triangles.

Derived Values

From area K, compute circumradius R, inradius r, and heights. Heron's formula opens the door to full triangle analysis.

History

Named after Hero of Alexandria (c. 10-70 AD). Also known as the Hero formula, it was known earlier to Archimedes.

Teaching Example: Triangle a=5, b=6, c=7. s = (5+6+7)/2 = 9. K = sqrt(9x4x3x2) = sqrt(216) = 14.697. R = (5x6x7)/(4x14.697) = 3.572. r = 14.697/9 = 1.633. Heights: h_a = 29.394/5 = 5.879, h_b = 29.394/6 = 4.899, h_c = 29.394/7 = 4.199.

Applications

Land Surveying Construction Navigation Math Education Engineering Competitions

FAQs about Heron's Formula

What is Heron's formula?▼

K = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. It calculates triangle area from side lengths alone.

Can Heron's formula handle right triangles?▼

Yes, it works perfectly for right triangles. For sides 3,4,5: s=6, K = sqrt(6x3x2x1) = sqrt(36) = 6, matching basexheight/2 = 3x4/2 = 6.

What is the semiperimeter used for?▼

The semiperimeter s appears in Heron formula, inradius formula (r=K/s), and in many triangle inequality problems.

How do you find triangle height from area?▼

Height h_a = 2K / a, where a is the base and K is the area from Heron. For a=5, K=14.697: h = 29.394/5 = 5.879.

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