Question

fine the area of each regular poygon.round to the nearest tenth.(!)12cm
its heptagon

Answer 1

12cm --> Is this the side of the heptagon or its perimeter or its apothem or something else?

Answer 2

The area of a regular polygon is one-half its apothem times its perimeter.

Answer 3

then how can i answer it?

Answer 4

I need to know what the 12 is you wrote in the problem.

Answer 5

there is 7 side in heptagon and 12 is one side of it.so how can i know the answer of this question?

Answer 6

Do you know the formuls for the apothem if you have the side?

Answer 7

@ PaxP --> I think trig will have to be used.

Answer 8

i think its (Area=1by2AP) right

Answer 9

... C is the center of polygon ... M side ... CM is the apothem

Answer 10

fine the area of the each polygon with the given vertices.
A(-1,-1),B(-2,3),C(2,4),D(4,-1)

Answer 11

\[\large \left| (x_1y_2 - x_2y_1) + (x_2y_3 - x_3y_2) + (x_3y_4 - x_4y_3) + (x_4y1 - x_1y_4) \over 2 \right|\]

Answer 12

thanks

Answer 13

\[(-3-2)+(-8-6)+(-2-16)+(-4-1)= -5-14-18-5=\Large -42\]

\[\Large Area = \left| -42 \over 2 \right| =21\] ... works if points are sorted clock-wise or counterclockwise

Answer 14

another way to write it would be \[Area=\frac{1}{2}\left[\left(x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{4}+x_{4}y_{1}\right)\right]-\frac{1}{2}\left[\left(y_{1}x_{2}+y_{2}x_{3}+y_{3}x_{4}+y_{4}x_{1}\right)\right]\]

Answer 15

M is the midpoint of side of length s... apothem CM has length a... n is the number of sides of a polygon
\[\Large \therefore \tan (x) = {a \over s/2}\]\[\Large \implies a= \frac s 2\tan(x)\]

x is half of each angle of the polygon\[x={180°(n-2)\over n} \div 2= \Large {90°(n-2)\over n}\]

So area of each regular heptagon with side 12cm is
\[\Large=\frac12 \times \frac{12}2\tan \left(90° \cdot 5 \over 7 \right)\times (12\cdot7)\]
\[\huge \approx 523.3 cm^2\]