Question

How do you find the sum of the interior angle measures of a convex 16 gon?

Answer 1

Sum of interior angles of a convex 16 sided-polygon
( hexakaidecagon ) would be `2520^@`.

Explanation:

Whatever the number of sides of a polygon,

the sum of its exterior angles is always `360^@`

Further, each pair of exterior angle and interior angle adds up to `180^@`

Hence in a polygon with `n` sides (or angles),

the sum of all the interior and exterior angles would be `180^@xxn`

and sum of interior angles would be `180^@xxn-360^@=180^@(n-2)`

Hence sum of interior angles of a convex 16-sided polygon would be `180^@xx(16-2)=180^@xx14=2520^@`.

Answer 2

Any polygon can be broken up into triangles by drawing in all the possible diagonals from one vertex.

You will find that the number of triangles is always 2 less than the number of sides.

The sum of the angles in any triangle is `180°`

Consider the following:

`"shape"color(white)(......)"sides"color(white)(......)"triangles"color(white)(......)"sum int angles"`

`"triangle"color(white)(......)"3"color(white)(...............)"1"color(white)(............)180xx1 = 180°`

`"quad"color(white)(..........)"4"color(white)(...............)"2"color(white)(............)180xx2 = 360°`

`"pentagon"color(white)(....)"5"color(white)(...............)"3"color(white)(..........)180xx3 = 540°`

`"n-sides"color(white)(.........)"n"color(white)(.........)(n-2)color(white)(......)180(n-2)`

`"16-sides"color(white)(......)"16"color(white)(...........)14color(white)(..........)180xx14 = 2520°`