Can you please help me find the answer

Question

Find the area of the figure.  Round to the nearest hundreth.

A=  in²

Answer 1

This is an octagon (8 sides), not a hexagon (6 sides).

At each side, you can construct an isoceles triangle with the center of the octagon as the vertex, and each vertex angle will measure 45 degrees. If you bisect the vertex angle, you will have 16 right triangles, each of which is a vertex angle of 22.5 degrees.

Using SOH-CAH-TOA, the base is 21*sin(22.5 degrees) and the height is 21*cos(22.5 degrees). Since there are 16 such triangles, 16 * (1/2) * base * height =1,247.34 square inches, approximately (to 2 decimal places of accuracy).

Answer 2

hexagon with 21 inches from center to vertex

what is the area?= about 661.5sqr3 square inches

= sum of 6 equilateral triangles each with sides = 21

each area = .5(21^2)sin60 = 441sqr3/4

times 6 = 661.5sqr3= about 1145.751609

= about 1,146 in^2

or

hexagon area = square area minus 4 right triangle corners

the hexagon is inscribed in a square with

square's sides = 42, square's area = (2(21)^2 = 4(441) 1764

then subtract 4 triangle areas, each = 1/2 x 21(21/sqr3)sin30

= about 1,191.64sqr3

close but different

mistake(s) above somewhere

or another method (akin to cheating, but very useful)

use the polygon area calculator on line = 1145.75 in^2

go with the equilateral triangle calculation, 1st method above

slight arithmetic mistake somewhere in the square method

the hexagon is not drawn to scale, that may cause some misleading area calculations. Even (Even?) worse is the drawing shows an octagon not a hexagon. Weird. redo all calculations for an 8 sided polygon

Area = 2129.34 if 21 is from center to vertex