The ratios of the measures of an interior angle to an exterior angle of a regular polygon is 4 to 1 How many sides does the polygon have?

Answer 1

Answer: 10 sidesInterior angle = (180 * (number_of_sides - 2)) / number_of_sides

Interior angle = 180 * (10-2) / 10 = 180 * 8 / 10 = 144

Exterior angle = 360 / number_of_sides = 360 / 10 = 36 degrees

Ratio (144:36 = 4:1)

Definitions:The exterior angle (e) is the angle created from the side and the imaginary line that would be if the adjacent side were to be extended beyond the vertices. The interior angle (i) plus the exterior angle (e) add up to 180 degrees, so i + e = 180.

In a normal polygon, the sum of the exterior angle of all vertices adds up to 360 degrees; the sides do not overlap or cross. In a regular polygon each external angle is the same, and will divide evenly into 360 degrees to give the number of sides (n): 360 / e = n.

Proof:Given that the ratio of the two angles is 4 to 1: i = 4e.

Now substitute: i + e = 180; 4e + e = 5e = 180.

e = 180 / 5 = 36 degrees.

i = 180 - e = 144 degrees.

360/e = 360/36 = 10 sides.

The interior angle of a regular normal polygon of n sides is: i = (n-2)*180/n.

For n=10: i = (10-2)*180/10 = 8*18 = 144 degrees, so it checks.

Other i to e ratios:1 to 2: 3 sides: triangle (trigon)

2 to 2: 4 sides: tetrahedron (tetragon, square)

3 to 2: 5 sides: pentahedron (pentagon)

4 to 2: 6 sides: hexahedron (hexagon)

5 to 2: 7 sides: heptahedron

6 to 2: 8 sides: octahedron (octagon)

7 to 2: 9 sides: enneahedron (nonogon)

8 to 2: 10 sides: decahedron (decagon)