Find the area of a regular #15#-gon whose each side is #14mm.#?
2 Answers
Area of a regular
Explanation:
Any regular polygon with
Let
Observe that
Using trigonometry
i.e.
and area of triangle is
=
and area of polygon is
Therefore area of the regular
=
=
Explanation:
The size of an interior angle of a regular 15-sided polygon can be calculated:
All regular polygons can be divided into isosceles triangles by joining the vertices to the centre. The base angles of the triangles are half of the interior angles of the polygon.
The angles of the triangles in a 15-sided polygon are
The area of a triangle is:
To be able to find the height, divide an isosceles triangle into 2 right-angled triangles.
The angles are
The height of the triangle is the side opposite the
Using trig, we have
The area of one of the 15 isosceles triangles is therefore:
There are 15 such triangles, so the area of the polygon is: