**Triangle Calculator with Side-Angle-Side (SAS)**

**How to Use a Triangle Calculator with Side-Angle-Side (SAS) Input**

Calculating the properties of a triangle when you know two sides and the included angle can seem complex. However, a **Triangle Calculator with Side-Angle-Side (SAS)** input simplifies this process, guiding you through each calculation step-by-step. This guide will explain how to use the SAS triangle calculator, provide insight into the key formulas it uses, and outline what each result means.

## Table of Contents

### What is the **Triangle Calculator with Side-Angle-Side (SAS)** Condition?

The SAS condition means you know:

- The lengths of two sides in the triangle.
- The measure of the angle between those two sides.

With these three values, you can calculate the missing side and other triangle properties such as the remaining angles, perimeter, area, and more.

### Getting Started with the Triangle Calculator

The Triangle Calculator accepts three inputs:

**Side A**: The length of the first known side.**Side B**: The length of the second known side.**Angle C**: The angle between sides A and B in degrees.

After entering these values and clicking “Calculate,” the calculator performs several calculations and displays results for the perimeter, area, remaining angles, and the third side (also known as the diagonal).

### Step-by-Step Calculation Breakdown

This calculator offers detailed, step-by-step results for each calculation, helping you understand the mathematical process behind the scenes. Here’s an overview of each step:

**Calculate the Third Side (Side C)**:

Using the **Law of Cosines**, the calculator finds Side C (the third side of the triangle) with the formula:

**C=√A ^{2}+B^{2}−2⋅A⋅B⋅cos(C)**

This formula calculates the length of the side opposite Angle C. For example, if Side A = 5, Side B = 7, and Angle C = 45°, the formula is:

**C=√5 ^{2}+7^{2}−2⋅5⋅7⋅cos(45°)**

**Calculate the Perimeter**:

The perimeter is the sum of all sides:

**Perimeter = A + B + C**

This result gives the total boundary length of the triangle.

**Calculate the Area**:

Using the formula for the area based on two sides and the included angle:

**Area=0.5⋅A⋅B⋅sin(C)**

For instance, if A = 5, B = 7, and C = 45°, the area calculation is:

**Area=0.5⋅5⋅7⋅sin(45°)**

This result gives the internal area of the triangle.

**Calculate Angle A**:

By applying the **Law of Sines**, the calculator determines Angle A:

**A=sin ^{−1}(A⋅sin(C)/C)**

This calculation finds the value of Angle A based on Side A, the calculated Side C, and Angle C.

**Calculate Angle B**:

The last angle, Angle B, is found by using the triangle’s internal angle sum (180°):

**B=180°−A−C**

This ensures the angles total to 180°, as required in any triangle.

### Example Calculation

Assume the following input values:

**Side A**= 5**Side B**= 7**Angle C**= 45°

#### Results

**Side C (Diagonal)**: 6.83 (calculated using the Law of Cosines).**Perimeter**: 18.83 (sum of Side A, Side B, and Side C).**Area**: 12.36 (using the formula for area with two sides and an included angle).**Angle A**: 35.26° (calculated using the Law of Sines).**Angle B**: 99.74° (calculated as 180° – Angle A – Angle C).

Each of these calculations is displayed in individual boxes with unique colors and hover effects to make results easy to read and visually appealing.

### Benefits of Using the **Triangle Calculator with Side-Angle-Side (SAS)**

This calculator is especially useful in:

**Education**: Students learning trigonometry can gain a deeper understanding of triangle properties and calculations.**Engineering**: For tasks involving triangular components, knowing precise calculations can be critical.**Real-World Applications**: Many real-world structures and objects incorporate triangles, making this tool useful for quick geometry calculations.

By using the Triangle Calculator with SAS input, you can calculate missing sides and angles quickly, with a clear explanation of each step. This not only saves time but also builds a stronger understanding of trigonometric principles.