SAS Triangle Calculator – Solve Two Sides & Included Angle

SAS Triangle Calculator

Solve triangle given two sides and the included angle.

In triangle geometry, SAS stands for Side–Angle–Side. This means you already know the lengths of two sides of a triangle and the angle between them. Even though you have three values, the triangle is not fully solved yet. The missing side, the remaining angles, and the area still need to be calculated.

The SAS Triangle Calculator helps you solve this type of triangle quickly and accurately. Instead of using long trigonometric formulas by hand, you simply enter the known values, and the calculator finds the rest. This tool is useful for students, teachers, engineers, architects, and anyone working with triangle measurements.

What the SAS Triangle Calculator Is

A Side–Angle–Side Triangle Solver

The SAS Triangle Calculator is an online geometry tool designed to solve triangles when:

  • Two sides are known

  • The angle between those sides is known

This is called an SAS triangle. The calculator uses the Law of Cosines to find the third side and then calculates the remaining angles and other triangle properties.

What the Calculator Can Find

The calculator can determine:

  • The third side of the triangle

  • The remaining two angles

  • The area of the triangle

  • The perimeter

  • The triangle type (acute, right, or obtuse)

All results are based on standard trigonometry and geometry rules.

How the SAS Triangle Calculator Works

Step 1: Enter Two Sides and the Included Angle

You begin by entering the values of side a, side b, and the angle C between them.

Step 2: Apply the Law of Cosines

The calculator uses the Law of Cosines to find the third side.

Step 3: Find Remaining Angles

Once all three sides are known, the calculator finds the remaining angles using trigonometric formulas.

Step 4: Display Full Triangle Solution

The final results include side lengths, angles, area, and perimeter.

This process removes the need for manual calculations.

Key Formulas Used

Law of Cosines (Finding the Third Side)

When sides a and b and the included angle C are known:

c^2 = a^2 + b^2 - 2ab\cos(C) c = \sqrt{a^2 + b^2 - 2ab\cos(C)}

Finding the Remaining Angles

After finding side c, the other angles can be calculated:

A = \arccos!\left(\frac{b^2 + c^2 - a^2}{2bc}\right) B = 180^\circ - (A + C)

Area Formula (Using Two Sides and Included Angle)

\text{Area} = \frac{1}{2} ab\sin(C)

Perimeter Formula

\text{Perimeter} = a + b + c

These formulas work for all triangle types.

Step-by-Step Example

Given Triangle Values

  • a = 8 units

  • b = 11 units

  • C = 50°

Step 1: Find Side c

c = \sqrt{8^2 + 11^2 - 2(8)(11)\cos(50^\circ)} c \approx 8.51 \text{ units}

Step 2: Find Angle A

A = \arccos!\left(\frac{11^2 + 8.51^2 - 8^2}{2(11)(8.51)}\right) A \approx 44.1^\circ

Step 3: Find Angle B

B = 180^\circ - (50^\circ + 44.1^\circ) B \approx 85.9^\circ

Step 4: Find the Area

\text{Area} = \frac{1}{2}(8)(11)\sin(50^\circ) \text{Area} \approx 33.8 \text{ square units}

The calculator performs all these steps instantly.

Features of the SAS Triangle Calculator

Accurate Law of Cosines Calculations

The calculator applies the correct trigonometric formula for precise results.

Solves All Triangle Types

It works for acute, right, and obtuse triangles.

Simple Input System

Only three values are needed to solve the triangle.

Instant Results

All triangle properties are calculated in seconds.

Uses and Applications

Education and Homework

Students use the calculator to solve SAS triangle problems and check their answers.

Engineering and Construction

Engineers and builders use SAS calculations for structural layouts and measurements.

Surveying and Design

Surveyors and designers use triangle geometry to measure distances and angles accurately.

Tips to Avoid Common Mistakes

Many users accidentally use an angle that is not between the two known sides. The Law of Cosines only works correctly when the angle is the included angle.

Another common mistake is forgetting to convert angle units. Always make sure the angle is in degrees, not radians, unless stated otherwise.

Some users round values too early, which can reduce accuracy in the final angles and area. Let the calculator complete all steps before rounding.

Using inconsistent measurement units, such as mixing meters and feet, can cause incorrect results. Always use the same unit system.

Finally, remember that triangle angles must add up to 180°. If they do not, the input values are invalid.

FAQs

What does SAS mean?

SAS stands for Side–Angle–Side.

Which formula is used for SAS triangles?

The Law of Cosines is used.

Can the calculator find the area?

Yes, it uses the sine area formula.

Does it work for all triangles?

Yes, as long as two sides and the included angle are known.

Final Words

The SAS Triangle Calculator is a powerful and easy-to-use geometry tool for solving triangles using the Side–Angle–Side method.

Whether you are a student, teacher, engineer, or surveyor, this calculator saves time, improves accuracy, and makes triangle problem-solving simple and reliable.

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