Question Find the area of a regular hexagon with side length $18$ .

Studdy Solution

STEP 1

1. A regular hexagon is a polygon with six equal sides and six equal angles.
2. The area of a regular hexagon can be calculated by dividing it into six equilateral triangles and finding the area of one of those triangles.
3. The formula for the area of an equilateral triangle with side length $s$ is $\frac{\sqrt{3}}{4}s^2$ .
4. The area of the hexagon will be six times the area of one of the equilateral triangles.

STEP 2

1. Calculate the area of one equilateral triangle with side length 18.
2. Multiply the area of the equilateral triangle by 6 to get the area of the regular hexagon.

STEP 3

Calculate the area of one equilateral triangle using the formula $\frac{\sqrt{3}}{4}s^2$ with $s=18$ .
$A_{\triangle} = \frac{\sqrt{3}}{4} \times 18^2$

STEP 4

Simplify the calculation to find the area of the equilateral triangle.
$A_{\triangle} = \frac{\sqrt{3}}{4} \times 324$

STEP 5

Further simplify the calculation by multiplying the numbers.
$A_{\triangle} = 81\sqrt{3}$

STEP 6

Multiply the area of the equilateral triangle by 6 to find the area of the regular hexagon.
$A_{\text{hexagon}} = 6 \times A_{\triangle}$

STEP 7

Substitute the area of the equilateral triangle into the equation and calculate the area of the hexagon.
$A_{\text{hexagon}} = 6 \times 81\sqrt{3}$

STEP 8

Simplify the calculation to get the final area of the hexagon.
$A_{\text{hexagon}} = 486\sqrt{3}$

STEP 9

Calculate the numerical value of $486\sqrt{3}$ to find the exact area.
$A_{\text{hexagon}} \approx 486 \times 1.732$

STEP 10

Perform the multiplication to get the final answer.
$A_{\text{hexagon}} \approx 842.4$
The area of the regular hexagon with a side length of 18 is approximately 842.4 square units. The correct answer from the given options is 842.4.

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