Triangle Area Calculator
Area, perimeter, and type from three sides via Heron.
What it does
Heron formula finds triangle area from three sides alone: halve the perimeter, multiply by each leftover, and root the product. No heights or angles needed, which suits surveying and construction tapes.
The tool above takes the three side lengths. It returns the area with perimeter and the triangle type beside them.
The default 3, 4, 5 gives semiperimeter 6 with root(6x3x2x1) = 6 area, perimeter 12, right scalene. Equilateral 5, 5, 5 reads 10.83 area with perimeter 15.
Heron algebra follows Khan Academy geometry units with the same type rules in LibreTexts math references. Both reject side sets failing the triangle inequality first.
How to use
- Enter the three side lengths in one unit.
- Check each sits below the other two summed.
- Read area first for the surface.
- Read perimeter for fencing and trim.
- Use type for the shape name.
Formula
- s
- Semiperimeter, half the fence.
- leftovers
- s minus each side in turn.
- root
- Square root closing the product.
Units square for area, plain for perimeter; mixing units corrupts both. Degenerate flat triples (a+b=c) fail the strict inequality and are rejected.
Worked example
Example: 3, 4, 5
Defaults: sides 3, 4, 5.
- Semiperimeter: (3+4+5)/2 = 6.
- Leftovers: 3, 2, 1.
- Product: 6 x 3 x 2 x 1 = 36.
- Area: root(36) = 6. Perimeter 12. Type right scalene.
- Tool shows 6, 12, Scalene, right.
Reference table
| Sides | Area | Type |
|---|---|---|
| 3, 4, 5 | 6 | Right scalene |
| 5, 5, 5 | 10.83 | Equilateral |
| 5, 5, 8 | 12 | Isosceles |
| 2, 3, 10 | rejected | Cannot close |
Limits
- Side lengths only; angles need trig pages.
- Strict inequality required, no flat triples.
- Sides cap at a billion units.
- Near degenerate shapes round delicately.
Common mistakes
- Forgetting to halve the perimeter first.
- Rooting before multiplying all four terms.
- Mixing meters with feet mid triple.
- Accepting 2, 3, 10 as a triangle.
- Reading area units as plain length.
Try it yourself
Quick checks. Answers sit tucked under each question.
5, 5, 5?
s 7.5, area root(117.19) = 10.83, equilateral.
5, 5, 8?
s 9, root(9x4x4x1) = 12, isosceles.
2, 3, 10?
Rejected: 2 + 3 sits below 10.
Questions
How do you find area from sides?
Heron: half perimeter s, then root of s(s-a)(s-b)(s-c).
What is 3-4-5 area?
6 square units with perimeter 12.
When is it right?
When squares add: 9 + 16 = 25 here.
Why reject 2, 3, 10?
Triangle inequality: each side must sit below the other two summed.
Equilateral area shortcut?
Root 3 over 4 times side squared.
Heron vs base height?
Same answer; Heron skips measuring heights.