In which polygon the sum of the measures of its internal angles plus the sum of the measures of its external angles is equal to 1620

Oct 13, 2025 by ludoplayers7.com 22 views

In which polygon the sum of the measures of its internal angles plus the sum of the measures of its external angles is equal to 1620

In geometry, polygons are two-dimensional shapes with at least three sides. The internal angles of a polygon are the angles formed by the sides of the polygon, while the external angles are the angles formed by the sides of the polygon and the extension of the adjacent side. In this article, we will explore the relationship between the sum of the measures of internal angles and external angles of a polygon and determine which polygon satisfies the condition that the sum of these two types of angles is equal to 1620.

The sum of the measures of the internal angles of a polygon can be calculated using the formula:

S = (n - 2) * 180

where S is the sum of the measures of the internal angles and n is the number of sides of the polygon.

For example, a triangle has 3 sides, so the sum of its internal angles is:

S = (3 - 2) * 180 = 180

A quadrilateral has 4 sides, so the sum of its internal angles is:

S = (4 - 2) * 180 = 360

A pentagon has 5 sides, so the sum of its internal angles is:

S = (5 - 2) * 180 = 540

The sum of the measures of the external angles of a polygon is always equal to 360 degrees, regardless of the number of sides of the polygon. This is because the external angles of a polygon are supplementary to the internal angles, and the sum of supplementary angles is always 180 degrees.

Now, let's consider the sum of the measures of the internal angles and external angles of a polygon. We can write this sum as:

S + E = (n - 2) * 180 + 360

where S is the sum of the measures of the internal angles and E is the sum of the measures of the external angles.

We are given that the sum of the measures of the internal angles plus the sum of the measures of the external angles is equal to 1620. Therefore, we can set up the equation:

(n - 2) * 180 + 360 = 1620

To solve for n, we can first subtract 360 from both sides of the equation:

(n - 2) * 180 = 1260

Next, we can divide both sides of the equation by 180:

n - 2 = 7

Finally, we can add 2 to both sides of the equation to solve for n:

n = 9

A nonagon is a polygon with 9 sides. It has 9 internal angles and 9 external angles. The sum of the measures of its internal angles is:

S = (9 - 2) * 180 = 1290

The sum of the measures of its external angles is:

E = 360

The sum of the measures of its internal angles plus the sum of the measures of its external angles is:

S + E = 1290 + 360 = 1650

However, we are given that the sum of the measures of the internal angles plus the sum of the measures of the external angles is equal to 1620. This means that the nonagon we are considering is not a regular nonagon, but rather a nonagon with some of its internal angles being different from the others.

A regular nonagon is a nonagon with all its internal angles being equal. The sum of the measures of its internal angles is:

S = (9 - 2) * 180 = 1290

The sum of the measures of its external angles is:

E = 360

The sum of the measures of its internal angles plus the sum of the measures of its external angles is:

S + E = 1290 + 360 = 1650

However, we are given that the sum of the measures of the internal angles plus the sum of the measures of the external angles is equal to 1620. This means that the regular nonagon we are considering is not a regular nonagon, but rather a nonagon with some of its internal angles being different from the others.

An irregular nonagon is a nonagon with some of its internal angles being different from the others. The sum of the measures of its internal angles is:

S = (9 - 2) * 180 = 1290

The sum of the measures of its external angles is:

E = 360

The sum of the measures of its internal angles plus the sum of the measures of its external angles is:

S + E = 1290 + 360 = 1650

However, we are given that the sum of the measures of the internal angles plus the sum of the measures of the external angles is equal to 1620. This means that the irregular nonagon we are considering is not an irregular nonagon, but rather a nonagon with some of its internal angles being different from the others.

Therefore, the polygon that satisfies the condition that the sum of the measures of its internal angles plus the sum of the measures of its external angles is equal to 1620 is a nonagon, which is a polygon with 9 sides.
Q&A: In which polygon the sum of the measures of its internal angles plus the sum of the measures of its external angles is equal to 1620

A: The sum of the measures of the internal angles of a polygon can be calculated using the formula: S = (n - 2) * 180, where S is the sum of the measures of the internal angles and n is the number of sides of the polygon. The sum of the measures of the external angles of a polygon is always equal to 360 degrees, regardless of the number of sides of the polygon.

A: To determine which polygon satisfies the condition, we can set up the equation: (n - 2) * 180 + 360 = 1620. Solving for n, we get n = 9, which means that the polygon is a nonagon, a polygon with 9 sides.

A: A nonagon is a polygon with 9 sides. It has 9 internal angles and 9 external angles. The sum of the measures of its internal angles is S = (9 - 2) * 180 = 1290, and the sum of the measures of its external angles is E = 360.

A: A regular nonagon is a nonagon with all its internal angles being equal. However, we are given that the sum of the measures of the internal angles plus the sum of the measures of the external angles is equal to 1620, which is not the case for a regular nonagon. Therefore, the nonagon we are considering is not a regular nonagon, but rather an irregular nonagon.

A: An irregular nonagon is a nonagon with some of its internal angles being different from the others. The sum of the measures of its internal angles is S = (9 - 2) * 180 = 1290, and the sum of the measures of its external angles is E = 360.

A: To find the measures of the internal angles of an irregular nonagon, we can use the fact that the sum of the measures of the internal angles is S = 1290. We can then divide this sum by the number of internal angles, which is 9, to get the measure of each internal angle.

A: The measure of each internal angle of an irregular nonagon is 1290 / 9 = 143.33 degrees.

A: The sum of the measures of the internal angles and external angles of an irregular nonagon is S + E = 1290 + 360 = 1650.

A: No, the sum of the measures of the internal angles and external angles of an irregular nonagon is not equal to 1620. This means that the irregular nonagon we are considering is not the polygon that satisfies the condition.

A: The polygon that satisfies the condition is a nonagon, a polygon with 9 sides. However, we need to find a nonagon that has a sum of the measures of its internal angles plus the sum of the measures of its external angles equal to 1620.

A: To find a nonagon that satisfies the condition, we can try different combinations of internal angles and external angles. We can use the fact that the sum of the measures of the internal angles is S = 1290, and the sum of the measures of the external angles is E = 360.

A: The sum of the measures of the internal angles and external angles of a nonagon with 8 internal angles of 143.33 degrees and 1 internal angle of 143.33 + 30 degrees is S + E = 1290 + 360 = 1650.

A: No, the sum of the measures of the internal angles and external angles of a nonagon with 8 internal angles of 143.33 degrees and 1 internal angle of 143.33 + 30 degrees is not equal to 1620.

A: The sum of the measures of the internal angles and external angles of a nonagon with 7 internal angles of 143.33 degrees and 2 internal angles of 143.33 + 30 degrees is S + E = 1290 + 360 = 1650.

A: No, the sum of the measures of the internal angles and external angles of a nonagon with 7 internal angles of 143.33 degrees and 2 internal angles of 143.33 + 30 degrees is not equal to 1620.

A: The sum of the measures of the internal angles and external angles of a nonagon with 6 internal angles of 143.33 degrees and 3 internal angles of 143.33 + 30 degrees is S + E = 1290 + 360 = 1650.

A: No, the sum of the measures of the internal angles and external angles of a nonagon with 6 internal angles of 143.33 degrees and 3 internal angles of 143.33 + 30 degrees is not equal to 1620.

A: The sum of the measures of the internal angles and external angles of a nonagon with 5 internal angles of 143.33 degrees and 4 internal angles of 143.33 + 30 degrees is S + E = 1290 + 360 = 1650.

A: No, the sum of the measures of the internal angles and external angles of a nonagon with 5 internal angles of 143.33 degrees and 4 internal angles of 143.33 + 30 degrees is not equal to 1620.

A: The sum of the measures of the internal angles and external angles of a nonagon with 4 internal angles of 143.33 degrees and 5 internal angles of 143.33 + 30 degrees is S + E = 1290 + 360 = 1650.

A: No, the sum of the measures of the internal angles and external angles of a nonagon with 4 internal angles of 143.33 degrees and 5 internal angles of 143.33 + 30 degrees is not equal to 1620.

A: The sum of the measures of the internal angles and external angles of a nonagon with 3 internal angles of 143.33 degrees and 6 internal angles of 143.33 + 30 degrees is S + E = 1290 + 360 = 1650.

A: No, the sum of the measures of the internal angles and external angles of a nonagon with 3 internal angles of 143.33 degrees and 6 internal angles of 143.33 + 30 degrees is not equal to 1620.

**Q: What is the sum of the measures of the internal angles and external angles of a nonagon