- Find the sum of the interior angles of a decagon. 2.If the exterior angle of a regular polygon is 30°, how many sides does the polygon have? 3.A regular polygon has an interior angle of 160°.How many sides does it have? 4.If the sum of the interior angles of a regular polygon is 1080°, how many sides does the polygon have?
- Find the sum of the interior angles of a decagon.
2.If the exterior angle of a regular polygon is 30°, how many sides does the polygon have?
3.A regular polygon has an interior angle of 160°.How many sides does it have?
4.If the sum of the interior angles of a regular polygon is 1080°, how many sides does the polygon have?
The Correct Answer and Explanation is :
Let’s break down and answer each of the four questions one by one.
1. Sum of the Interior Angles of a Decagon
A decagon has 10 sides. To find the sum of the interior angles of any polygon, we use the formula:
[
\text{Sum of interior angles} = (n – 2) \times 180^\circ
]
where (n) is the number of sides. For a decagon ((n = 10)):
[
\text{Sum of interior angles} = (10 – 2) \times 180^\circ = 8 \times 180^\circ = 1440^\circ
]
So, the sum of the interior angles of a decagon is 1440°.
2. If the Exterior Angle of a Regular Polygon is 30°
For any regular polygon, the exterior angle is given by:
[
\text{Exterior angle} = \frac{360^\circ}{n}
]
where (n) is the number of sides. If the exterior angle is (30^\circ), we set up the equation:
[
30^\circ = \frac{360^\circ}{n}
]
Solving for (n):
[
n = \frac{360^\circ}{30^\circ} = 12
]
Thus, the polygon has 12 sides.
3. If the Interior Angle of a Regular Polygon is 160°
The formula for the interior angle of a regular polygon is:
[
\text{Interior angle} = \frac{(n – 2) \times 180^\circ}{n}
]
We are given that the interior angle is (160^\circ). Setting up the equation:
[
160^\circ = \frac{(n – 2) \times 180^\circ}{n}
]
Multiplying both sides by (n):
[
160n = (n – 2) \times 180
]
Expanding:
[
160n = 180n – 360
]
Rearranging the terms:
[
180n – 160n = 360
]
[
20n = 360
]
Solving for (n):
[
n = \frac{360}{20} = 18
]
Thus, the polygon has 18 sides.
4. If the Sum of the Interior Angles of a Regular Polygon is 1080°
The sum of the interior angles of a regular polygon is given by:
[
\text{Sum of interior angles} = (n – 2) \times 180^\circ
]
We are given that the sum is 1080°. Setting up the equation:
[
1080^\circ = (n – 2) \times 180^\circ
]
Dividing both sides by 180°:
[
\frac{1080^\circ}{180^\circ} = n – 2
]
[
6 = n – 2
]
Solving for (n):
[
n = 6 + 2 = 8
]
Thus, the polygon has 8 sides.
Summary of Answers:
- The sum of the interior angles of a decagon is 1440°.
- A regular polygon with an exterior angle of 30° has 12 sides.
- A regular polygon with an interior angle of 160° has 18 sides.
- A regular polygon with a sum of interior angles of 1080° has 8 sides.
These problems involve applying formulas for polygons, which are essential for understanding the geometric properties of various shapes.