in order to find the area of a regular hexagon using the formula Area=1/2(nxs)(a)you must frist find the apothem whie vertical leg of a 30-60-90 triangle with the base of 3. Find the length of the apothem in the 30-60-90 triangle
tan30 = a/3,
a = 3tan30 = 1.732.
To find the length of the apothem in a 30-60-90 triangle, we can use the ratios of the sides in this special triangle. In a 30-60-90 triangle, the ratios of the side lengths are:
- The ratio of the short leg to the hypotenuse is 1:2.
- The ratio of the long leg to the hypotenuse is √3:2.
We are given the base of the triangle, which is the short leg with a length of 3. The apothem of a regular hexagon is perpendicular to one side and intersects the center of the hexagon, so it acts as the height in the 30-60-90 triangle.
Since the ratio of the short leg to the hypotenuse is 1:2, we can set up the following proportion:
3/h = 1/2
To solve for h, the hypotenuse, we can cross multiply:
2 * 3 = h
h = 6
Now, we know the length of the hypotenuse (6) and the short leg (3). To find the length of the apothem, which is the height in the 30-60-90 triangle, we can use the ratio of the long leg to the hypotenuse:
(long leg)/h = √3/2
Substituting the known values:
(long leg)/6 = √3/2
To isolate the long leg, cross multiply:
2 * (long leg) = 6 * √3
2 * (long leg) = 6√3
(long leg) = 3√3
Therefore, the length of the apothem in the 30-60-90 triangle is 3√3.