Triangle Area Calculator by 3 Sides | Heron's Formula
Calculate triangle area using Heron's formula with three sides. Enter side lengths to get exact area instantly. Perfect for geometry and math homework.
Visualization
Formula
Area = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2Heron's formula calculates the area of a triangle when all three side lengths are known. It's particularly useful when the height is difficult to measure directly.
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Use Cases
- •Construction and architecture projects
- •Land surveying and property measurement
- •Engineering calculations and design
- •Mathematical problem solving and education
Frequently Asked Questions
Common questions about calculating triangle area using three sides (Heron's formula).
What is Heron's formula and how does it work?
Heron's formula calculates the area of a triangle when you know all three side lengths. The formula is: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2 is the semi-perimeter. It's named after the ancient Greek mathematician Heron of Alexandria.
When should I use Heron's formula?
Use Heron's formula when you know the lengths of all three sides of a triangle but not the height. This is common in surveying, construction, and when working with triangular plots of land.
What makes a valid triangle for the three sides method?
A triangle is valid if the sum of any two sides is greater than the third side. This is called the triangle inequality theorem. For example, sides 3, 4, 5 form a valid triangle because 3+4>5, 3+5>4, and 4+5>3.
Can Heron's formula be used for any type of triangle?
Yes! Heron's formula works for all types of triangles - acute, obtuse, right, equilateral, isosceles, and scalene triangles. The formula doesn't depend on the angles, only on the side lengths.
How accurate is Heron's formula?
Heron's formula is mathematically exact when the side lengths are precise. However, the accuracy of your result depends on the precision of your input measurements. Small errors in side measurements can lead to larger errors in the calculated area.
Detailed Explanation
Learn more about Heron's formula and its applications.
Historical Background
Heron's formula is named after Heron of Alexandria (also known as Hero), a Greek mathematician and engineer who lived in the 1st century AD. Although the formula is attributed to Heron, it was actually known to Archimedes centuries earlier.
Mathematical Derivation
The formula is derived from the Law of Cosines and the standard area formula. Starting with the standard area formula A = (1/2)bh and using the Law of Cosines to express the height in terms of the sides, we arrive at Heron's formula.
Accuracy and Precision
Heron's formula is mathematically exact. However, in practical applications, the accuracy depends on the precision of your measurements. For high-precision applications, consider the coordinate method which can be more accurate for digital calculations.