Angle Between 3 Points in Regular Polygon Calculator

Angle Between 3 Vertices Equation Lesson Note

The Angle between 3 vertices equation solver is a valuable tool for calculating the angle between three points in a regular polygon with n sides. This equation solver is particularly useful when three vertices (a, b, and c) are given, and the angles subtended at vertex a1, a2, and a3 need to be determined. To illustrate its application, let’s consider two worked examples:

Example 1:
Suppose we have a regular hexagon with vertices labeled as A, B, C, D, E, and F. We are interested in calculating the angle subtended at vertex B, given that the vertices A, B, and C are provided.

Using the Angle between 3 vertices equation solver, we can solve for the angle at vertex B as follows:

1. Assign coordinates to the vertices:
A(0, 0), B(1, 0), C(0.5, √3/2) (Note: these coordinates form an equilateral triangle)

2. Calculate the slopes of the lines:
Slope of AB = (0 – 0) / (1 – 0) = 0
Slope of BC = (√3/2 – 0) / (0.5 – 1) = -√3

3. Use the slope formula to calculate the angle:
Angle at vertex B = arctan((Slope of BC – Slope of AB) / (1 + Slope of BC * Slope of AB))
= arctan((-√3 – 0) / (1 + (-√3 * 0)))
= arctan(-√3 / 1)
= -60 degrees

Therefore, the angle subtended at vertex B in this regular hexagon is -60 degrees.

Example 2:
Consider a regular pentagon with vertices labeled as P, Q, R, S, and T. We want to determine the angle at vertex Q, given vertices P, Q, and R.

Following the steps of the Angle between 3 vertices equation solver:

1. Assign coordinates to the vertices:
P(0, 0), Q(1, 0), R(0.5, √5/2) (These coordinates form an isosceles triangle)

2. Calculate the slopes of the lines:
Slope of PQ = (0 – 0) / (1 – 0) = 0
Slope of QR = (√5/2 – 0) / (0.5 – 1) = -√5

3. Use the slope formula to calculate the angle:
Angle at vertex Q = arctan((Slope of QR – Slope of PQ) / (1 + Slope of QR * Slope of PQ))
= arctan((-√5 – 0) / (1 + (-√5 * 0)))
= arctan(-√5 / 1)
= -67.38 degrees (rounded to two decimal places)

Hence, the angle subtended at vertex Q in this regular pentagon is approximately -67.38 degrees.

In both examples, the Angle between 3 vertices equation solver proved effective in calculating the angles between three given points in regular polygons.