Calculating the Sum of Interior Angles in a 14-Sided Polygon

The sum of the interior angles in a polygon with 14 sides can be calculated using a well-established geometric formula. This article explains the computation step-by-step, including various methods and applications of the formula.

Sum of Interior Angles Formula

The sum of the interior angles in a polygon can be determined by the formula:

( text{Sum of interior angles} (n - 2) times 180^{circ} )

Where ( n ) is the number of sides in the polygon.

Using the Formula to Find the Sum for a 14-Sided Polygon

For a polygon that has 14 sides:

Substitute ( n ) with 14 in the formula: ( text{Sum of interior angles} (14 - 2) times 180^{circ} ) Simplify the expression: ( text{Sum of interior angles} 12 times 180^{circ} ) Perform the multiplication: ( text{Sum of interior angles} 2160^{circ} )

The sum of the interior angles in a 14-sided polygon is ( 2160^{circ} ).

Additional Methods to Find the Sum of Interior Angles

Other methods include:

Using the Interior and Exterior Angle Relationship

The measure of an exterior angle of a regular polygon can be calculated as:

( text{Exterior angle} frac{360^{circ}}{n} )

For a polygon with 14 sides:

( text{Exterior angle} frac{360^{circ}}{14} ) Simplify to find the exterior angle: ( text{Exterior angle} frac{180^{circ}}{7} approx 25.71428^{circ} ) The interior angle is found by subtracting the exterior angle from 180°: ( text{Interior angle} 180^{circ} - frac{180^{circ}}{7} approx 154.28572^{circ} ) Calculate the sum of the interior angles by multiplying the interior angle by the number of sides: ( 14 times text{Interior angle} approx 14 times 154.28572^{circ} 2160^{circ} )

Dividing the Polygon into Triangles

Another method is to divide the polygon into triangles by drawing lines from any interior point to the vertices:

Each polygon can be divided into ( n - 2 ) triangles. For a 14-sided polygon, this results in 12 triangles. The sum of the angles in each of the 12 triangles is ( 180^{circ} ). Thus, the total sum of the angles is: ( 12 times 180^{circ} 2160^{circ} ) The angles at the interior point contribute ( 360^{circ} ). The sum of the interior angles is then: ( 2160^{circ} - 360^{circ} 2160^{circ} )

Conclusion

Thus, the sum of the interior angles of a 14-sided polygon is ( 2160^{circ} ). This calculation is essential in various geometric applications and helps in understanding the properties of polygons.