Triangle area formulas (base-height, Heron, angle, coordinates)

Triangles are the “atoms” of geometry: any polygon can be split into triangles, and many real-world measurement problems boil down to triangle area. The challenge is that you don’t always have a convenient base and vertical height. That’s why this Triangle Area Calculator supports four practical methods, and automatically checks your inputs for validity.

If you know a base b and its perpendicular height h (the shortest distance from the base to the opposite vertex), the area is:

• A = (b × h) / 2

This is the simplest method and works for any triangle where height is measured at a right angle to the chosen base. If the height falls outside the triangle (obtuse triangles), the formula still works as long as h is perpendicular to the base line.

When you have all three side lengths a, b, and c, you can use Heron’s formula:

• Compute the semiperimeter: s = (a + b + c) / 2
• Then area: A = √( s(s−a)(s−b)(s−c) )

Heron’s formula is extremely useful for surveying, construction, and any situation where the height is hard to measure. The key requirement is the triangle inequality: each side must be shorter than the sum of the other two. If your numbers fail that test, the calculator will flag it as “not a valid triangle.”

If you know two sides and the angle between them (the included angle), you can compute area using sine:

• A = (1/2)ab sin(C) where C is the included angle between sides a and b

This is common in physics, engineering, and navigation because angles are often measured directly. The area grows as the sine of the angle: for fixed sides, the triangle has maximum area at 90°.

If your triangle vertices are points (x₁,y₁), (x₂,y₂), (x₃,y₃), area comes from a determinant (a “shoelace” style formula):

• A = 1/2 × | x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂) |

This is ideal for coordinate geometry, CAD sketches, map problems, and any time you have points rather than side lengths. The absolute value ensures the result is positive whether you list points clockwise or counterclockwise.

Validation, method switching, and unit-friendly output

To make this tool fast and reliable, it follows three steps:

• Step 1 — pick a method: You select the method that matches what you know (base-height, three sides, etc.).
• Step 2 — validate inputs: The calculator rejects missing values, negative numbers, and invalid triangles (triangle inequality for side-based methods).
• Step 3 — compute + explain: It calculates the area and prints the exact formula used, plus intermediate values (like semiperimeter for Heron’s formula).

You can also choose a unit label (cm, m, in, ft, etc.). The calculator doesn’t convert units automatically (because mixing units is ambiguous); instead it treats your inputs as consistent. If you enter base in meters, height must be in meters too, and the output becomes square meters (m²).

Real examples you can copy/paste

Base b = 10, height h = 6.

• A = (10 × 6) / 2 = 60 / 2 = 30

So the triangle area is 30 square units.

Sides a = 7, b = 8, c = 9.

• s = (7 + 8 + 9) / 2 = 24 / 2 = 12
• A = √(12(12−7)(12−8)(12−9))
• A = √(12 × 5 × 4 × 3) = √(720) ≈ 26.8328

So the area is about 26.83 square units.

Sides a = 9 and b = 5, included angle C = 30°.

• A = 1/2 × 9 × 5 × sin(30°)
• sin(30°) = 0.5
• A = 0.5 × 45 × 0.5 = 11.25

So the area is 11.25 square units.

Points (0,0), (6,0), (2,5).

• A = 1/2 × | 0(0−5) + 6(5−0) + 2(0−0) |
• A = 1/2 × | 0 + 30 + 0 | = 15

So the area is 15 square units. (You can visualize this as base 6 and height 5, then divide by 2.)

Getting the “right” triangle area every time

• Height must be perpendicular: If you measure “height” along a slanted edge, base-height will be wrong. Use Heron or the sine formula instead.
• Check triangle inequality: If a + b ≤ c, you don’t have a triangle. You have a flat line (area = 0) or impossible measurements.
• Angles must be included: The sine formula requires the angle between the two sides you enter, not a random triangle angle.
• Keep units consistent: Don’t mix meters and centimeters unless you convert first.
• Rounding: For Heron’s formula, rounding sides too aggressively can change the area noticeably. Use the most precise measurements you have.

Triangle Area Calculator FAQ

• Which method should I use?

  If you have a true perpendicular height, base-height is fastest. If you only have side lengths, Heron’s formula is perfect. If you measured an angle between two sides, use the sine method. If you’re working on a coordinate grid or map, use coordinates.

• What if my triangle is obtuse?

  All methods here work for obtuse triangles. For base-height, the height can fall outside the triangle, but it must still be perpendicular to the base line. If that’s confusing, use Heron’s formula or the sine method instead.

• Can triangle area ever be negative?

  No — area is always non-negative. In coordinate formulas, a signed area can be negative depending on point order (clockwise vs counterclockwise), but taking the absolute value gives the true geometric area.

• What does “square units” mean?

  Area is measured in squared units. If you input in meters, the result is square meters (m²). If you input in inches, the result is square inches (in²). The calculator lets you label units so your output reads naturally.

• Why does Heron’s formula sometimes show an error?

  It usually means your side lengths can’t form a triangle (triangle inequality fails), or one of the inputs is missing/negative. Fix the numbers and try again.

• How accurate is this calculator?

  The math is exact, but accuracy depends on your inputs. Measuring mistakes (especially height and angle) are the biggest source of error. If precision matters, measure multiple times and use consistent units.

This calculator is for educational and measurement support. Always verify critical construction or engineering measurements with professional tools.