Calculate the area of any triangle using 4 methods: base × height, Heron's formula (3 sides), SAS (2 sides + included angle), or coordinate geometry (3 vertices). Includes classification, inradius...
The area of a triangle is one of the most fundamental calculations in all of geometry. Whether you are a student solving homework problems, an engineer computing cross-sections, a surveyor measuring land parcels, or a programmer implementing collision detection, knowing how to compute triangle area efficiently is essential.
There are four primary methods for finding the area of a triangle.
**Base × Height** is the simplest: A = ½bh, where b is any side chosen as the base and h is the perpendicular distance from the base to the opposite vertex. This method is ideal when the base and height are directly measurable.
**Heron's Formula** uses all three side lengths: first compute the semi-perimeter s = (a + b + c)/2, then A = √[s(s − a)(s − b)(s − c)]. This is invaluable when you know the sides but not the height.
**SAS (Side-Angle-Side)** requires two sides and the included angle: A = ½ab sin(C). Surveyors and navigators use this method frequently because angles are often easier to measure in the field than perpendicular heights.
**Coordinate Method** takes three vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) and computes A = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. This is the go-to method in computer graphics, GIS, and computational geometry.
This calculator supports all four methods, automatically classifies the triangle (equilateral, isosceles, scalene; acute, right, obtuse), and computes the inradius and circumradius when side lengths are available.
This triangle area calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Method, Unit, Base, Height and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Base × Height: A = ½ × b × h Heron's: s = (a+b+c)/2, A = √[s(s−a)(s−b)(s−c)] SAS: A = ½ × a × b × sin(C) Coordinates: A = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| Inradius: r = A / s Circumradius: R = (abc) / (4A)
Result: For method=base-height, base=10, height=6, the tool returns the solved triangle area outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in triangle area formulas and reports derived values, checks, and classifications automatically.
This page is tailored to triangle area, with outputs tied directly to the form fields (Method, Unit, Base, Height). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
A = ½ × base × height is the simplest. Pick any side as the base, measure the perpendicular height from that side to the opposite vertex, and multiply.
Heron's formula computes the area from three sides: first find the semi-perimeter s = (a+b+c)/2, then A = √[s(s−a)(s−b)(s−c)]. It works for any valid triangle.
Use A = ½ab sin(C) when you know two sides and the included angle. This is common in surveying and navigation where angle measurements are more practical than height measurements.
Use the shoelace formula: A = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|. Plug in the (x,y) coordinates of all three vertices.
The inradius (r) is the radius of the inscribed circle (largest circle fitting inside the triangle). The circumradius (R) is the radius of the circumscribed circle (circle passing through all three vertices).
Technically no — if the three points are collinear (all on one line), the "triangle" is degenerate and has zero area. A valid triangle always has positive area.