Triangle Area Calculator
Calculate triangle area using base and height, Heron's three-sides formula, coordinate geometry, or the SAS trigonometric method.
Choose a shape
Triangle inputs
Master area formula library
Rectangle
A = l × w
Length times width.
Square
A = s²
A rectangle with equal sides.
Triangle
A = ½ × b × h
Half the base times the height.
Triangle (Heron)
A = √[s(s−a)(s−b)(s−c)]
From three side lengths only.
Trapezoid
A = ½(b₁ + b₂) × h
Average of the parallel bases × height.
Circle
A = π × r²
Pi times the radius squared.
Sector
A = ½ × r² × θ
Radians for θ.
Ellipse
A = π × a × b
Semi-major × semi-minor × π.
Parallelogram
A = b × h
Base times perpendicular height.
Rhombus
A = ½ × d₁ × d₂
Half the product of the diagonals.
Kite
A = ½ × d₁ × d₂
Same as the rhombus diagonal rule.
Regular polygon
A = ½ × n × s × a
n sides of length s, apothem a.
Polygon
A = ½ |Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|
Shoelace / Surveyor's formula.
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What Is the Triangle Area Formula?
The triangle is the simplest closed polygon — three sides, three vertices, three angles summing to 180°. Despite that simplicity, triangles are the building blocks of every other polygon, the basis of trigonometry, and the workhorse of structural engineering. Four different formulas reach the same area depending on which inputs you have.
The formula used on this page is A = ½ × b × h or A = √[s(s−a)(s−b)(s−c)]. The calculator applies it instantly, returns full unit conversions, and shows every substitution in the step-by-step panel.
How the Calculation Works
Four area formulas matter. (1) Base × height: A = ½bh — pick any side as the base and drop a perpendicular from the opposite vertex. (2) Heron's formula: from three side lengths a, b, c, compute s = (a + b + c)/2 then A = √[s(s − a)(s − b)(s − c)]. (3) Side-Angle-Side trigonometry: A = ½ × a × b × sin C, where C is the included angle. (4) Shoelace formula: A = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)| from three vertex coordinates. Behind the scenes the calculator picks the appropriate method and falls back to the law of cosines whenever it needs to recover the other two angles.
6 Ways to Use This Triangle Calculator
Compute classroom area problems
Whether you have the base and height, three sides, or three vertex coordinates, the calculator picks the right formula and shows the worked steps.
Solve land surveyor traverses
Triangulated parcels are decomposed into triangles, and each is solved with Heron's formula from the three measured sides.
Decompose irregular shapes
Any polygon can be split into triangles. Sum their areas to find the total, no matter how irregular the outline.
Identify triangle types
The calculator classifies the result as acute, right, or obtuse and as scalene, isosceles, or equilateral so you can sanity-check shape definitions.
Find the area from a CAD outline
Paste three (x, y) vertices and the Shoelace formula returns the area directly — useful for technical drawing and engineering checks.
Recover the third angle
Once two sides and an included angle are known, the law of cosines fills in the third side, then the law of sines or law of cosines finishes the other two angles.
Best Practices
- Triangle inequality. Any single side must be shorter than the sum of the other two. If a + b ≤ c, no triangle exists and Heron's formula returns a non-real number.
- Wrong height definition. The 'height' in A = ½bh is perpendicular to the chosen base. Using the slanted side as the height overestimates the area.
- Degree-radian mix-ups. The SAS formula uses sin C — make sure C is in the angle unit your trig function expects.
- Coordinate order. The Shoelace formula returns a signed area that depends on vertex order; take the absolute value to get the geometric area.
Why Triangle Area Matters
Structural engineering
Trusses, gables, and bracing rely on rigid triangles; a triangle is the only polygon that does not deform under fixed-length sides.
Surveying and GIS
Plots are triangulated into elementary triangles whose areas are summed for the total parcel area.
Computer graphics
Triangles are the universal primitive — every 3D mesh is rendered as a triangle soup, and each face's area shapes lighting and texture sampling.
Navigation triangulation
Finding position from three known points (GPS, radio direction-finding) is geometrically a triangle-area problem.
Common Mistakes
- Using slant height in the base × height formula. Drop a perpendicular from the apex to the base line; that perpendicular is the height.
- Forgetting absolute value in the Shoelace formula. Signed area can be negative depending on vertex direction. Take |·| at the end.
- Misclassifying isosceles vs equilateral. Equilateral requires all three sides equal; isosceles requires only two.
Core Formula
Formula
A = ½ × b × h or A = √[s(s−a)(s−b)(s−c)]
Area is always returned in your input unit squared. The conversion panel translates the result into mm², cm², m², ha, km², in², ft², yd², acres, and mi² so you can compare across unit systems.
Methodology & Accuracy
Every calculation runs entirely in your browser using JavaScript's full double-precision arithmetic (about 15–17 significant digits internally). The displayed result is rounded to four significant figures by default; numbers outside the 0.0001 to 999,999,999 range switch automatically to scientific notation. The triangle formula used here is the standard Euclidean-geometry identity, sourced from the same textbooks used in secondary-school and university geometry courses.
Frequently Asked Questions
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