Triangle Side Angle Calculator

Calculate the unknown triangle side, angles, perimeter, area, semiperimeter, inradius, and circumradius using two sides and their included angle. Triangle Side Angle Calculator helps students, engineers, builders, and surveyors solve practical triangle measurements accurately.

Highlights

  • Calculate the third triangle side using two known sides and their included angle.
  • Solve all three triangle angles with accurate trigonometric calculations.
  • Find the triangle’s area, perimeter, and semiperimeter instantly.
  • Calculate the inradius and circumradius for advanced geometric analysis.
  • Supports multiple length and angle units, including meters, feet, degrees, radians, and more.

Table of Contents

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

How Triangle Side Angle Calculator Works?

The Triangle Side Angle Calculator uses the side-side-angle (SAS) relationship: two known sides, a and b, and their included angle, C. It first finds the unknown side c with the Law of Cosines.

c=a2+b2−2abcos(C)

After calculating the third side, the calculator determines the remaining angles. Angle A, opposite side a, and angle B, opposite side b, can be calculated using the Law of Cosines.

A=arccos(b2+c2−a22bc),B=arccos(a2+c2−b22ac)

The supplied included angle is displayed as angle C, and the three angles satisfy A + B + C = 180° when expressed in degrees. The perimeter and semiperimeter are calculated from the three sides:

P=a+b+c,s=P2

The area uses the two known sides and their included angle, while the inradius and circumradius use standard triangle relationships.

K=12absin(C),r=Ks,R=c2sin(C)

Here, K is the triangle area, r is the inradius, and R is the circumradius. The calculator accepts the listed length formats for the side inputs and outputs, and degree, radian, or gon formats for the angular values.

Practical Use Cases

Students can enter two sides and their included angle to check trigonometry assignments, verify Law of Cosines calculations, and explore how changing an angle affects the third side, area, and remaining angles.

Engineers can use the calculator for preliminary triangular geometry, component layouts, force-diagram dimensions, and quick checks of perimeter, area, inradius, and circumradius before applying a detailed design workflow.

Builders and fabricators can determine an unknown diagonal or cut length from two measured lengths and the angle between them, then use the perimeter and area results for planning triangular frames, panels, and other structures.

Surveyors can solve a triangle from two observed distances and an included angle, obtaining the third distance and the complete set of angles for field-check calculations and site-layout verification.

Designers and makers can calculate triangular surface area and boundary length for material estimates, while the inradius and circumradius support circular fitting and geometric construction tasks.

Accuracy and Limitations

The calculator uses standard double-precision trigonometric and geometric formulas, so results are generally reliable for valid positive side lengths and an included angle strictly between 0 and 180 degrees. Accuracy can decrease for extremely small or extremely large side values, angles very close to 0 or 180 degrees, or nearly degenerate triangles because the area and circumradius calculations become sensitive to rounding. The inputs are limited to 0.000001 through 1,000,000,000 for each side and 0.000001 through 179.999999 for the included angle. Results are rounded for display, so retain more precision when using them in later calculations.

Quick Examples

Example #1: Right Triangle Survey

Calculate the missing side and geometric properties of a triangle with two measured sides and a right included angle.

Example #2: Construction Triangle

Determine triangle dimensions for a construction layout using centimeter measurements and an included angle.

Example #3: Land Measurement Triangle

Estimate triangle sides and area for a land-surveying application using feet and gradians.

Frequently Asked Questions

The included angle is the angle formed by two known sides. In triangle notation, it is the angle located between sides a and b.

The three interior angles of every triangle add up to 180 degrees, or π radians. Knowing two angles is enough to determine the third.

Perimeter is the total distance around a triangle, while area measures the amount of space enclosed by its sides. Perimeter uses linear units, and area uses square units.

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.

Triangle measurements are commonly used in construction, surveying, engineering, architecture, navigation, land measurement, and classroom geometry.

Cite This Calculator