ABC Triangle Calculator

Calculate angles, area, perimeter, heights, medians, radii, and other triangle properties based on three side lengths. ABC Triangle Calculator helps students, engineers, architects, and geometry professionals analyze triangle dimensions accurately.

Highlights

  • Calculate all three triangle angles from the side lengths using the Law of Cosines.
  • Find area, perimeter, and semiperimeter with accurate geometric formulas.
  • Determine heights and medians for each side of the triangle.
  • Calculate inradius, circumradius, and all three exradii.
  • Support multiple length units, including metric and imperial measurements.

Table of Contents

  1. How ABC Triangle Calculator Works?
  2. Practical Use Cases
  3. Accuracy and Limitations
  4. Quick Examples
  5. Frequently Asked Questions (FAQs)
  6. Cite This Calculator

How ABC Triangle Calculator Works?

The ABC Triangle Calculator determines the complete geometry of a triangle from three positive side lengths: a, b, and c. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The three sides must satisfy the triangle inequality: each side must be less than the sum of the other two sides.

a<b+c,b<a+c,c<a+b

First, the semiperimeter is calculated as half of the perimeter. The area is then found with Heron’s formula, which works directly from the three side lengths.

s=a+b+c2,K=s(s-a)(s-b)(s-c)

The three interior angles are calculated with the Law of Cosines. The calculator can display these angles in degrees or radians.

A=cos-1(b2+c2-a22bc)
B=cos-1(a2+c2-b22ac),C=cos-1(a2+b2-c22ab)

Each altitude follows from the area relationship K = bh/2. Therefore, the heights to sides a, b, and c are calculated as 2K/a, 2K/b, and 2K/c. The medians use Apollonius’s theorem:

ma=2b2+2c2-a22,mb=2a2+2c2-b22,mc=2a2+2b2-c22

The inradius, circumradius, and three exradii are derived from the area and semiperimeter.


OutputFormulaGeometric meaning
PerimeterP = a + b + cTotal boundary length
Inradiusr = K/sRadius of the inscribed circle
CircumradiusR = abc/(4K)Radius of the circle through all vertices
Exradius opposite Ara = K/(s − a)Radius of the excircle opposite angle A
Exradius opposite Brb = K/(s − b)Radius of the excircle opposite angle B
Exradius opposite Crc = K/(s − c)Radius of the excircle opposite angle C

The inputs support millimeters, centimeters, meters, kilometers, inches, feet, yards, and miles. Length outputs use the selected length basis, area uses the corresponding square-area basis, and angles are available in degrees or radians.

Practical Use Cases

Students and teachers: Geometry learners can enter three known side lengths to verify the triangle inequality, explore Heron’s formula, check the Law of Cosines, and compare calculated angles, heights, medians, and radii with classroom solutions.

Engineers: Engineers can use the calculator to evaluate triangular members, panels, supports, and layouts when all three side dimensions are known. The perimeter, altitudes, medians, circumradius, and exradii provide useful preliminary geometry for design checks.

Architects and designers: Architects can analyze triangular floor plans, roof sections, façade elements, and site layouts. The area and perimeter support early quantity estimates, while the angles and heights help define the shape accurately.

Surveyors and mapping professionals: Surveyors can use three measured triangle sides to obtain the associated interior angles, area, and radii for basic triangulation and geometric reference work.

Woodworkers and fabricators: Fabricators can determine corner angles, perpendicular heights, and median locations for triangular panels, braces, templates, and cut plans from three side measurements.

Geometry and CAD professionals: CAD users can use the calculated dimensions as a quick independent check when constructing triangular features, locating midpoints, or validating a model’s basic proportions.

Accuracy and Limitations

The calculator uses standard double-precision numerical evaluation of the Law of Cosines, Heron’s formula, altitude and median relationships, and radius formulas, so results are generally reliable for valid positive side lengths within the allowed input range of 0.000001 to 1,000,000,000. The three sides must satisfy all triangle inequalities; otherwise, no real triangle exists and area-based results are invalid. Very small areas, nearly degenerate triangles, extremely uneven side magnitudes, or values close to the input limits can increase floating-point rounding effects, particularly in Heron’s formula and inverse-cosine angle calculations. Displayed results should therefore be rounded appropriately for the precision of the original measurements and should not replace a professional structural, surveying, or engineering review.

Quick Examples

Example #1: Standard Metric Triangle

Analyze a moderate-sized triangle using centimeter measurements.

Example #2: Right Triangle

Calculate properties of a right triangle with a 3-4-5 side relationship.

Example #3: Large Isosceles Triangle

Evaluate an isosceles triangle using large-scale measurements in feet.

Frequently Asked Questions

The sum of any two sides must be greater than the third side. This is known as the triangle inequality.

The largest angle is opposite the longest side, while the smallest angle is opposite the shortest side. All three interior angles add up to 180°.

A median connects a vertex to the midpoint of the opposite side. A height, or altitude, is the perpendicular distance from a vertex to the opposite side or its extension.

The inradius is the radius of the circle inside the triangle that touches all three sides. The circumradius is the radius of the circle passing through all three vertices.

Students, engineers, architects, surveyors, designers, and construction professionals use triangle measurements for planning, structural analysis, layouts, and distance calculations.

Cite This Calculator