Acute Triangle Calculator
Calculate the angles, area, perimeter, radii, and altitudes based on three side lengths. Acute Triangle Calculator helps students, educators, engineers, and geometry enthusiasts analyze valid acute triangles.
Highlights
- Checks whether three side lengths form a valid acute triangle.
- Calculates all three triangle angles using the law of cosines.
- Finds perimeter, semiperimeter, and area using Heron’s formula.
- Determines the altitudes corresponding to each triangle side.
- Calculates the triangle’s circumradius and inradius.
Table of Contents
How Acute Triangle Calculator Works?
The calculator uses three side lengths, a, b, and c, to determine whether they form a valid acute triangle and then calculates its angles, perimeter, semiperimeter, area, altitudes, circumradius, and inradius.
First, the side lengths must satisfy the triangle inequality:
After sorting the sides so that c is the largest, the triangle is acute when the square of the largest side is less than the sum of the squares of the other two sides:
The three angles are found with the law of cosines. The angle opposite each side is calculated as follows:
Angles can be displayed in degrees or radians. The perimeter and semiperimeter are:
Heron's formula calculates the area from the three sides:
Each altitude follows from the area relation K = ½bh. The radii are calculated using the area and side lengths:
| Output | Calculation |
|---|---|
| Perimeter | a + b + c |
| Semiperimeter | (a + b + c) / 2 |
| Area | Heron's formula |
| Altitude to a side | 2K / side |
| Circumradius | abc / 4K |
| Inradius | K / s |
Practical Use Cases
Students can enter three known side lengths to check whether a triangle is acute, practice the law of cosines and Heron's formula, and compare calculated angles, area, and altitudes with textbook exercises.
Educators can use the calculator to create examples involving valid, invalid, acute, and non-acute triangles, while demonstrating how side relationships affect angles, radii, and height measurements.
Engineers and designers can estimate the geometric properties of triangular components from side measurements, including the enclosed area, perimeter, vertex-circle radius, inscribed-circle radius, and perpendicular heights.
Surveyors and construction planners can analyze a triangle defined by three measured sides and obtain its angles and altitudes for layout checks or preliminary geometry calculations.
Geometry enthusiasts can investigate how changing one side affects the triangle's classification, angle structure, area, and related circle measurements.
Accuracy and Limitations
The calculator uses standard double-precision numerical evaluations of the law of cosines, Heron's formula, and the radius and altitude relationships, so results are generally reliable for ordinary valid inputs within the allowed range of 0.01 to 1,000,000,000 for each side. It requires strict triangle inequalities and specifically identifies an acute triangle using the squared-side test; degenerate, impossible, right, or obtuse triangles are not suitable for acute-triangle results. Very large sides with extremely small differences can lose precision because of floating-point rounding, and rounded displayed angles or measurements may not reproduce the exact input values when recalculated manually.
Quick Examples
Example #1: Equilateral Triangle
Analyze a regular acute triangle with three equal side lengths.
Example #2: Scalene Triangle
Calculate the properties of a valid acute triangle with three different side lengths.
Example #3: Large Acute Triangle
Evaluate angles, area, radii, and altitudes for a larger acute triangle.
