Question

name each of the following regular polygons. determine the sum of interior angles and the sum of exterior angles.
1)
type
2)
type:
3)
type
4)
type.
5)
type:
6)
type:
7)
type:
8)
type:
9)
type:
10)
type
11)
type.
12)
type:

Answer:

To solve this, we'll analyze each regular polygon by counting its sides, naming it, then using the formulas for the sum of interior angles (\(S = (n - 2) \times 180^\circ\)) and the sum of exterior angles (always \(360^\circ\) for any convex polygon).

1)

• Step 1: Count Sides

The polygon has 12 sides.

• Step 2: Name the Polygon

A 12 - sided polygon is a dodecagon.

• Step 3: Sum of Interior Angles

Using \(S=(n - 2)\times180^\circ\) with \(n = 12\):
\(S=(12 - 2)\times180^\circ=10\times180^\circ = 1800^\circ\).

• Step 4: Sum of Exterior Angles

For any convex polygon, the sum of exterior angles is \(360^\circ\).

2)

• Step 1: Count Sides

The polygon has 7 sides.

• Step 2: Name the Polygon

A 7 - sided polygon is a heptagon (or septagon).

• Step 3: Sum of Interior Angles

Using \(S=(n - 2)\times180^\circ\) with \(n = 7\):
\(S=(7 - 2)\times180^\circ=5\times180^\circ = 900^\circ\).

• Step 4: Sum of Exterior Angles

Sum of exterior angles is \(360^\circ\).

3)

• Step 1: Count Sides

The polygon has 10 sides.

• Step 2: Name the Polygon

A 10 - sided polygon is a decagon.

• Step 3: Sum of Interior Angles

Using \(S=(n - 2)\times180^\circ\) with \(n = 10\):
\(S=(10 - 2)\times180^\circ=8\times180^\circ = 1440^\circ\).

• Step 4: Sum of Exterior Angles

Sum of exterior angles is \(360^\circ\).

4)

• Step 1: Count Sides

The polygon has 9 sides.

• Step 2: Name the Polygon

A 9 - sided polygon is a nonagon (or enneagon).

• Step 3: Sum of Interior Angles

Using \(S=(n - 2)\times180^\circ\) with \(n = 9\):
\(S=(9 - 2)\times180^\circ=7\times180^\circ = 1260^\circ\).

• Step 4: Sum of Exterior Angles

Sum of exterior angles is \(360^\circ\).

5)

• Step 1: Count Sides

The polygon has 5 sides.

• Step 2: Name the Polygon

A 5 - sided polygon is a pentagon.

• Step 3: Sum of Interior Angles

Using \(S=(n - 2)\times180^\circ\) with \(n = 5\):
\(S=(5 - 2)\times180^\circ=3\times180^\circ = 540^\circ\).

• Step 4: Sum of Exterior Angles

Sum of exterior angles is \(360^\circ\).

6)

• Step 1: Count Sides

The polygon has 5 sides.

• Step 2: Name the Polygon

A 5 - sided polygon is a pentagon.

• Step 3: Sum of Interior Angles

Using \(S=(n - 2)\times180^\circ\) with \(n = 5\):
\(S=(5 - 2)\times180^\circ=3\times180^\circ = 540^\circ\).

• Step 4: Sum of Exterior Angles

Sum of exterior angles is \(360^\circ\).

7)

• Step 1: Count Sides

The polygon has 7 sides.

• Step 2: Name the Polygon

A 7 - sided polygon is a heptagon (or septagon).

• Step 3: Sum of Interior Angles

Using \(S=(n - 2)\times180^\circ\) with \(n = 7\):
\(S=(7 - 2)\times180^\circ=5\times180^\circ = 900^\circ\).

• Step 4: Sum of Exterior Angles

Sum of exterior angles is \(360^\circ\).

8)

• Step 1: Count Sides

The polygon has 11 sides.

• Step 2: Name the Polygon

An 11 - sided polygon is a hendecagon (or undecagon).

• Step 3: Sum of Interior Angles

Using \(S=(n - 2)\times180^\circ\) with \(n = 11\):
\(S=(11 - 2)\times180^\circ=9\times180^\circ = 1620^\circ\).

• Step 4: Sum of Exterior Angles

Sum of exterior angles is \(360^\circ\).

9)

• Step 1: Count Sides

The polygon has 9 sides.

• Step 2: Name the Polygon

A 9 - sided polygon is a nonagon (or enneagon).

• Step 3: Sum of Interior Angles

Using \(S=(n - 2)\times180^\circ\) with \(n = 9\):
\(S=(9 - 2)\times180^\circ=7\times180^\circ = 1260^\circ\).

• Step 4: Sum of Exterior Angles

Sum of exterior angles is \(360^\circ\).

10)

• Step 1: Count Sides

The polygon has 10 sides.

• Step 2: Name the Polygon

A 10 - sided polygon is a decagon.

• Step 3: Sum of Interior Angles

Using \(S=(n - 2)\times180^\circ\) with \(n = 10\):
\(S=(10 - 2)\times180^\circ=8\times180^\circ = 1440^\circ\).

• Step 4: Sum of Exterior Angles

Sum of exterior angles is \(360^\circ\).

11)

• Step 1: Count Sides

The polygon has 12 sides.

• Step 2: Name the Polygon

A 12 - sided polygon is a dodecagon.

• Step 3: Sum of Interior Angles

Using \(S=(n - 2)\times180^\circ\) with \(n = 12\):
\(S=(12 - 2)\times180^\circ=10\times180^\circ = 1800^\circ\).

• Step 4: Sum of Exterior Angles

Sum of exterior angles is \(360^\circ\).

12)

• Step 1: Count Sides

The polygon has 8 sides.

• Step 2: Name the Polygon

An 8 - sided polygon is an octagon.

• Step 3: Sum of Interior Angles

Using \(S=(n - 2)\times180^\circ\) with \(n = 8\):
\(S=(8 - 2)\times180^\circ=6\times180^\circ = 1080^\circ\).

• Step 4: Sum of Exterior Angles

Sum of exterior angles is \(360^\circ\).

Final Answers (Summarized)

Polygon #	Type	Sum of Interior Angles	Sum of Exterior Angles
2)	Heptagon	\(900^\circ\)	\(360^\circ\)
3)	Decagon	\(1440^\circ\)	\(360^\circ\)
4)	Nonagon	\(1260^\circ\)	\(360^\circ\)
5)	Pentagon	\(540^\circ\)	\(360^\circ\)
6)	Pentagon	\(540^\circ\)	\(360^\circ\)
7)	Heptagon	\(900^\circ\)	\(360^\circ\)
8)	Hendecagon	\(1620^\circ\)	\(360^\circ\)
9)	Nonagon	\(1260^\circ\)	\(360^\circ\)
10)	Decagon	\(1440^\circ\)	\(360^\circ\)
11)	Dodecagon	\(1800^\circ\)	\(360^\circ\)
12)	Octagon	\(1080^\circ\)	\(360^\circ\)

(Note: For each polygon, the "Type" is based on the number of sides, and the sum of exterior angles is always \(360^\circ\) for convex polygons.)