## Triangle Calculator with Angle-Side-Angle (ASA)

### How to Use the Triangle Calculator with Angle-Side-Angle (ASA)

This Triangle Calculator with Angle-Side-Angle (ASA), where you know two angles and one side. By entering your known values, the calculator quickly computes the remaining angles, sides, and other properties of the triangle. Here’s a step-by-step guide on how to use it effectively.

## Table of Contents

#### Step 1: Enter Your Known Values

**Input the First Angle**:

- In the calculator, enter the measurement of one of the given angles, let’s call it
**Angle B**. - Ensure that the angle is in degrees. If your angle is in radians, you’ll need to convert it to degrees.

**Input the Second Angle**:

- Now, enter the second known angle, called
**Angle C**. - The calculator automatically sums the angles to ensure they are consistent with a valid triangle. The sum of angles in any triangle must be exactly 180°.

**Input the Known Side**:

- Finally, input the length of the side opposite one of the given angles. Let’s call this side
**a**. - This is essential for the calculator to use the
**Law of Sines**to find the other sides.

#### Step 2: Calculate the Missing Third Angle

- Once you enter two angles, the calculator finds the missing
**Angle A**by subtracting the sum of the known angles from 180°. - This step is essential as it provides the final angle, completing the triangle’s angle measurements.

#### Step 3: Calculate the Remaining Sides Using the Law of Sines

The calculator applies the **Law of Sines** to calculate the lengths of the missing sides ( b ) and ( c ).

Using the formula:

**a/sin (A) = b/sin (B) = c/sin (C)**

It finds side ( b ) by rearranging:

**b=a⋅sin (B)/sin (A)**

Similarly, it finds side ( c ) as:

**c=a⋅sin(C)/sin(A)**

#### Step 4: Calculate Additional Triangle Properties

**Perimeter**:

Once all sides ( a ), ( b ), and ( c ) are known, the perimeter ( P ) is calculated as:

**P = a + b + c**

**Area**:

The area of the triangle is calculated using the Side-Angle-Side formula:

**Area=1/2 x a x b x sin(C)**

**Medians**:

The length of each median (a line from a vertex to the midpoint of the opposite side) is calculated. For example:

**Median from A=1/2 √2b ^{2}+2c^{2}−a^{2}**

**Heights**:

The calculator provides the height from each vertex to its opposite side, calculated as:

**Height from A=2 x Area** / **a**

**Inradius and Circumradius**:

**Inradius** ( r ) (the radius of the inscribed circle) is calculated as:

**r = Area **/ **semi-perimeter**

**Circumradius** ( R ) (radius of the circumscribed circle) is calculated as:

**R = a** / **2 sin (A)**

**Centroid**:

- The calculator locates the centroid, which is the intersection point of the medians. If coordinate inputs are available, the centroid is found by averaging the coordinates of the vertices.

#### Step 5: Review Results

**Detailed Output**:- The calculator displays all results in a clear, organized manner. You’ll see values for:
**All three angles****All three side lengths****Perimeter and area****Medians and heights****Inradius, circumradius, and centroid**

**Graphical Representation**(if available):- Some calculators may provide a visual representation of the triangle based on your inputs, helping you better understand the calculated values in a real-world context.

### Example for Triangle Calculator with Angle-Side-Angle (ASA)

Suppose you know:

**Angle B**= 45°**Angle C**= 60°**Side a**(opposite Angle A) = 10 cm

Here’s how the calculator works:

**Calculates Angle A**:

**A = 180° – (45° + 60°) = 75°**

**Calculates Side b**:

**b = 10⋅sin(45°) / sin(75°) ≈ 7.32 cm**

**Calculates Side c**:

**c = 0⋅sin(60°) / sin(75°)1 ≈ 8.66 cm**

**Calculates Perimeter**:

**P = 10 + 7.32 + 8.66 = 25.98 cm**

**Calculates Area**:

**Area = 1/2 ⋅ 10 ⋅ 7.32 ⋅ sin(60°) ≈ 31.75 cm2**

The calculator handles these calculations efficiently, giving accurate results for all triangle properties based on the initial Triangle Calculator with Angle-Side-Angle (ASA).