What is the measure of the point angle for a star-shaped polygon with 8 points and a dent angle of 130 degrees?
How do you solve this?
To find the measure of the point angle for a star-shaped polygon with 8 points and a dent angle of 130 degrees, you can use the following steps:
Step 1: Determine the total sum of the internal angles of the polygon.
For any polygon with n sides, the formula to calculate the total sum of internal angles is given by:
Total sum of internal angles = (n - 2) * 180 degrees.
For an 8-sided polygon, the total sum of internal angles would be:
(8 - 2) * 180 = 6 * 180 = 1080 degrees.
Step 2: Determine the number of dents in the polygon.
In this case, the polygon has 8 points and 8 sides. Since each dent corresponds to one side, the number of dents is the same as the number of sides, which is 8.
Step 3: Calculate the sum of the dent angles.
Since each dent angle is given as 130 degrees, the sum of the dent angles would be:
Sum of dent angles = number of dents * dent angle = 8 * 130 = 1040 degrees.
Step 4: Calculate the measure of the point angle.
To find the measure of the point angle, subtract the sum of the dent angles from the total sum of internal angles:
Measure of point angle = total sum of internal angles - sum of dent angles.
Measure of point angle = 1080 - 1040 = 40 degrees.
Therefore, the measure of the point angle for the star-shaped polygon with 8 points and a dent angle of 130 degrees is 40 degrees.
To find the measure of the point angle for a star-shaped polygon with 8 points and a dent angle of 130 degrees, you can follow these steps:
Step 1: Determine the total angle at the center of the polygon.
In a star-shaped polygon, the total angle at the center is equal to 360 degrees.
Step 2: Calculate the sum of the dent angles.
The sum of the dent angles is equal to the total angle at the center minus the sum of the point angles. In this case, the sum of the dent angles can be calculated as (8 - 2) * 180 degrees, since the polygon has 8 points (edges) and 8 - 2 = 6 triangles are formed. Therefore, the sum of the dent angles is 6 * 180 degrees = 1080 degrees.
Step 3: Find the measure of each dent angle.
Since the polygon has 8 points, there are 8 dent angles. To find the measure of each dent angle, divide the sum of the dent angles by the number of dent angles. In this case, 1080 degrees divided by 8 equals 135 degrees.
Step 4: Calculate the measure of the point angle.
The point angle is defined as the supplement of the dent angle. In other words, it is the difference between 180 degrees and the dent angle. In this case, the measure of the point angle is 180 degrees - 135 degrees = 45 degrees.
Therefore, the measure of the point angle for a star-shaped polygon with 8 points and a dent angle of 130 degrees is 45 degrees.