A regular polygon with 12 sides (dodecagon) is inscribed in a square of area 24 square units as shown in the figure where four of the vertices are mid points of the sides of the square . The area of the dodecagon in square units is. ans-19.26
Answers
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Answer:
area of dodecagon is 18 square units.
Hello. Hope this helps.
Step-by-step explanation:
Let r be half the side of the square.
Consider one of the twelve triangle segments made by a side of the dodecagon and the centre of the dodecagon ( = the centre of the square).
The angle at the centre is 360° / 12 = 30°.
The two sides of the triangle that meet at the centre each have length r, as this is the radius of the circumscribed circle.
The area of this small triangle is then (1/2) r² sin 30° = r² / 4.
The dodecagon is made up of twelve of these triangles, so
area(dodecagon) = 12 × r² / 4 = 3 r².
As r is half the side of the square,
area(square) = 4 r².
Therefore
area(dodecagon ) / area(square) = 3 / 4
=> area(dodecagon) = 3 × 24 / 4 = 3 × 6 = 18