Find teh area of a regular hexagon if its inscribed circle has a radius of 6in?
Okay so the question is, is the hexagon in the circle or is the circle inside the hexagon?
2007-05-30 13:45:25 · 9 answers · asked by Liz 2 in Science & Mathematics ➔ Mathematics
Hi,
The circle is in the hexagon, so its radius of 6 is the apothem of the hexagon. If you divide the hexagon into 6 equilateral triangles and drop an altitude from the center of the hexagon to the midpoint of any side of the hexagon, that forms two 30-60-90 triangles. That apothem of 6 inches id\s the side across from the 60 degree angle, so the 30 degree angle's opposite side would be 6/√3 or 2√3 inches long. That is half the length of the side of the hexagon so its side is 4√3 inches long. That means the perimeter is 6 times that, or 24√3 inches long.
The area of the hexagon can be found by multiplying:
Area = 1/2 x perimeter of the hexagon x apothem.
This becomes:
Area = 1/2 x 24√3 x 6 = 72√3 square inches.
This is the hexagon's area.
I hope that helps!! :-)
2007-05-30 13:55:39 · answer #1 · answered by Pi R Squared 7 · 0⤊ 0⤋
The inscribed circle of a hexagon is inside the hexagon.
So in this case the circle is inside. A regular hexagon can be divided into 6 congruent equilateral triangles. The radius of the circle is the height of one of the triangles.
h = height triangle = 6
s = side triangle
A = area triangle
6A = area hexagon
s = (2/√3)h
A = (1/2)sh = (1/2)(2/√3)hh = h²/√3
6A = 6h²/√3 = 6*6²/√3 = 216/√3 = 72√3
2007-05-30 13:55:00 · answer #2 · answered by Northstar 7 · 0⤊ 0⤋
Since it is an inscribed circle, the circle is inside the hexagon.
2007-05-30 13:49:03 · answer #3 · answered by smui0123 3 · 0⤊ 0⤋
the circle is NOT the regular hexagion.
the formula to find the area of a regular polygon when there is an inscribed circle is:
A = r*s
s = simi-perimeter (or P/2) of the regular polygon
r = radius of the circle
using trig, the side length of the regula hexagion is 12/sqrt(3)
A = 6 * (6 * 12/sqrt(3)/2 )
A = 124.71in^2
2007-05-30 14:01:05 · answer #4 · answered by 7 · 0⤊ 0⤋
Circle is in the hexagon.
2007-05-30 13:48:36 · answer #5 · answered by pathak_surendra 2 · 0⤊ 0⤋
Inscribed means inside, the circle is inside. circumscribed the other.
2007-05-30 13:52:32 · answer #6 · answered by Diane L 1 · 0⤊ 0⤋
inscribed circle = circle inside hexagon
circumscribed circle = hexagram inside circle
2007-05-30 13:49:15 · answer #7 · answered by Alex 4 · 0⤊ 0⤋
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2016-12-12 06:56:59 · answer #8 · answered by kreitman 4 · 0⤊ 0⤋
It says 'inscribed circle' so...........
2007-05-30 13:50:18 · answer #9 · answered by Anonymous · 0⤊ 0⤋