Question : ABCDEF is a regular hexagon. The side of the hexagon is 36 cm. What is the area of the $\triangle ABC$?
Option 1: $324 \sqrt{3}\ {cm}^2$
Option 2: $360 \sqrt{3} \ {cm}^2$
Option 3: $240 \sqrt{3} \ {cm}^2$
Option 4: $192 \sqrt{3} \ {cm}^2$
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Correct Answer: $324 \sqrt{3}\ {cm}^2$
Solution : Given: ABCDEF is a regular hexagon. The side of the hexagon is 36 cm. The area of the regular hexagon of side $a$ is $6\times \frac{\sqrt3}{4}a^2$. A regular hexagon is divided in to six equal triangles. So, the area of the $\triangle ABC$ $=\frac{1}{6}\times 6\times \frac{\sqrt3}{4}a^2$. ⇒ $\frac{1}{6}\times 6\times \frac{\sqrt3}{4}\times (36)^2=324 \sqrt3 \ cm^2$ Hence, the correct answer is $324\sqrt3 \ cm^2$.
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Question : ABCDEF is a regular hexagon of side 12 cm. What is the area (in cm2) of the $\triangle $ECD?
Question : $\triangle ABC$ is an equilateral triangle with a side of 12 cm and AD is the median. Find the length of GD if G is the centroid of $\triangle ABC$.
Question : In a triangle ${ABC}, {AB}={AC}$ and the perimeter of $\triangle {ABC}$ is $8(2+\sqrt{2}) $ cm. If the length of ${BC}$ is $\sqrt{2}$ times the length of ${AB}$, then find the area of $\triangle {ABC}$.
Question : ABC is an equilateral triangle. If the area of the triangle is $36 \sqrt{3}$, then what is the radius of the circle circumscribing the $\triangle ABC$?
Question : The base of a right pyramid is an equilateral triangle with a side of 8 cm, and its height is $30 \sqrt{3}$ cm. The volume (in cm$^3$ ) of the pyramid is:
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