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 √2b2+2c2−a2
- 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 = 25.98Â 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).