If a regular hexagon is inscribed in a circle with a radius of 4, the

Regular hexagon will have six traingle regions. Central angle will be 60. (360/6)
Equilateral triangle will be formed as we have three 60 angles. Length of each side is equal to radius 4.
Area of hexagon = area of 6 eq. triangles = 6 * root 3/4 * 4^2 = 24 root3

In an regular hexagon inscribed in a circle, its side is equal the radius.
We can divide the hexagon in 6 triangles each with the base of 4. The heigth will equal \(\sqrt{4^2-2^2}=\sqrt{12}=2\sqrt{3}\). To obtain this just use Pythagoras, the hypotenuse of each triangle it's the radius, and the bases it's \(\frac{4}{2}=2\).

Now you have the height of each triangle, so \(A_t=(4*2\sqrt{3})/2=4\sqrt{3}\).

\(A_h=6*A_t=6*4\sqrt{3}=24\sqrt{3}\)

A regular hexagon is essentially composed of 6 equilateral trianlges...and the line joining the opposite vertices is the diameter of the circle in which the hexagon is inscribed...So the radius of the circle forms the side of the equilateral triangle...
Area is 6* (3^1/2)/4(a^2) where a = radius of the circle.
6*sq. rt3/4 * 4^2=24 root 3

Questions about hexagons:
a-regular-hexagon-has-a-perimeter-of-30-units-what-is-the-131591.html
regular-hexagon-abcdef-has-a-perimeter-of-36-o-is-the-cente-89544.html

Hope it helps.
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The Hexagon can be divided into 6 equilateral triangles , each with side = radius of the circle. Since area of equilateral triangle is (root(3)*a^2)4, for 6 such triangles we will get 24root3

1) Each of the hexagon's angles 120 degrees : formula if you don't know it is [(# of sides - 2) x 180] = (6 - 2) x 180 = 720 ÷ 6 = 120.
2) Next, split up the hexagon into 6 equilateral triangles with 4 for each of its sides.
3) Find the area of one of the triangles:
- Base = 4
- find the height by splitting the triangle in half so that it becomes a 30/60/90 triangle and find the height using the pythagorean theorem or knowing the 1,√3,2 triangle. Height = 2√3
- one triangle's area = (1/2)bh = (1/2)(4)(2√3) = 4√3
4) find the area of the hexagon by multiplying the one triangle's area by 6:
- 6 x 4√3 = 24√3
Answer is D

Great method we could use the formula for the area of equilateral triangle and save a few steps Root 3/4 a^2 where a = 4

the hexagon comprises 6 identical equilateral triangles
area of 1 equilateral triangle=√3/4*4^2=4√3
6*4√3=24√3
D

We may recall that a regular hexagon can be divided into 6 equilateral triangles, and thus the area of a regular hexagon is:

6[(s^2√3)/4], where s = side of the equilateral triangle and (s^2√3)/4 = the area of the equilateral triangle.
Since the radius of 4 also represents one side of the equilateral triangle, we can now determine the area of the hexagon.

Area = 6[(4^2√3)/4]

Area = 6[(16√3)/4]

Area = 6(4√3)

Area = 24√3

Answer: D
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Will the length of the side of a regular polygon (not just a hexagon) inscribed in a circle always be equal to the radius?

No. Consider a square, or an equilateral triangle inscribed in a circle. Do their sides equal to the radius?
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In regular hexagon,the length of a diagonal is equal to two times the length of the side, so diagonal is 8 and each side (a) is 4.

Area of hexagon = square root 3*3*a2/2= 24*square root 3.

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area of hexagon = 3 * √3/4 * s^2
s=4
answer IMO D; . \(24 \sqrt{3}\)

For a hexagon inscribed in a circle, the radius of the circle is equal to the side of the hexagon. Therefore, in this situation, side of hexagon is 4. Formula for area of hexagon is ((3*square-root 3)/2)*a^2. Put a=4.
area of hexagon= (3*square-root 3*4^2)/ 2= 24 square-root 3

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