This triangle calculator finds area from three side lengths with Heron’s formula. Enter sides a, b, and c. The default 3, 4, 5 triangle has semi-perimeter 6 and area 6.
Use it when you know all three sides but not the height. For a right triangle hypotenuse from two legs, try the right triangle calculator. For the Pythagorean missing-side path, use the Pythagorean theorem calculator.
How the formula works
First form s = (a + b + c) ÷ 2. Then area = √[s(s – a)(s – b)(s – c)]. The triangle inequality must hold: each side shorter than the sum of the other two.
Worked example
a = 3, b = 4, c = 5. Semi-perimeter s = 6. Area = √[6×3×2×1] = √36 = 6.
| Input | Value |
|---|---|
| Side a | 3 |
| Side b | 4 |
| Side c | 5 |
| Area | 6 |
How to use the fields
- Side a, side b, and side c are the three positive lengths of the triangle.
Why Heron helps
Base and height are not always easy to measure on site. Three edge lengths often are. Heron turns those lengths into area without placing an altitude first.
Common mistakes
- Entering sides that violate the triangle inequality
- Using perimeter a+b+c as if it were area
- Mixing units across the three sides
- Expecting angle outputs from a sides-only tool
Right triangle shortcut
When you already know the triangle is right angled and the legs are the perpendicular sides, area is also (1/2)×leg1×leg2. For 3 and 4 that is again 6. Heron still works and confirms the same number.
Land and craft use
Plot corners as three measured edges when a triangular bed, sail panel, or yard section is irregular. Keep all tape measurements in one unit before you type them in.
Checking inequality quickly
For 3, 4, 5: 3+4>5, 3+5>4, and 4+5>3 all hold. For a bad set like 1, 2, 3, the sum 1+2 equals 3, so no area exists. Test inequality before trusting a result.
Classroom practice
Have students compute s on paper, then each (s – side) term, then the product under the root. Matching the calculator builds trust in every intermediate line.
Compare an equilateral example next: three equal sides should produce a familiar area formula. Keep the published default at 3-4-5 so shared screenshots stay consistent.
Scaling
If every side doubles, area grows by four. That square scaling matters for material estimates. Run Heron on the scaled sides rather than doubling the old area by mistake.
Similarity preserves shape ratios; area ratios are the square of the side ratio.
Height free field work
On uneven ground, dropping a perpendicular height can be awkward. Measuring three edges with a tape is often easier. Heron turns those edges into area without forcing a height first.
Keep the tape readings in one unit and avoid rounding each edge before the semi-perimeter step.
Obtuse and acute alike
Heron does not care whether the triangle is acute or obtuse as long as the sides form a valid triangle. That flexibility is why three side inputs are enough for a single area number.
Right triangles are included as a special case that can also be checked with (1/2)×leg×leg when you know the right angle location.
Material sheets
Fabric panels and sheet metal triangles start from area for purchasing, then add seam allowance separately. Do not bake seam allowance into side lengths unless your pattern says to.
Record side lengths next to the area so a cutter can verify the panel later.
Semi-perimeter discipline
Write s before any (s – side) term. Skipping the semi-perimeter and jumping into a memorized product is how people miscopy Heron. For 3-4-5, s is 6, and the four factors 6, 3, 2, and 1 multiply cleanly to 36 under the root.
If any (s – side) term is zero or negative, the sides do not form a proper triangle.
Limitations
Only side lengths are inputs. SSA, ASA, and angle chasing need other methods. Degenerate or impossible side sets produce no valid area.