Recall the formula to find the area of an equilateral triangle. You can also use the properties of 30^(∘)-60^(∘)-90^(∘) triangles, or alternatively, use the trigonometric ratios to find the apothem.
See solution.
Practice makes perfect
Let's consider a regular hexagon for which we only know the side length.
Next, we will describe three different ways to find the area of the hexagon above.
Using Equilateral Triangles
We can divide the hexagon into 6 equilateral triangles of side s.
Next, we will recall the area of an equilateral triangle of side length s.
A_T = s^2sqrt(3)/4
Finally, to find the area of the hexagon, we multiply the area of the triangle by 6.
A_H &= 6* A_T
&= 3 s^2sqrt(3)/2
Using a 30^(∘)-60^(∘)-90^(∘) Triangle
Let's mark a radius of the hexagon and one apothem.
We know the apothem bisects the central angle and also bisects each side of the polygon. From this, we get a 30^(∘)-60^(∘)-90^(∘) triangle whose shorter leg has a length of s/2. Now we can find the length of the longer leg.
a = s/2* sqrt(3)
Finally, we find the area of the hexagon by multiplying the apothem by the perimeter and dividing the result by 2.
ccl
A_H = 1/2* a* P
& ⇒ &
A_H = 1/2* ssqrt(3)/2 * 6 s [0.3cm]
& & A_H = 3 s^2sqrt(3)/2
Using Trigonometric Ratios
Let's mark a radius of the hexagon and one apothem.
As before, we get a 30^(∘)-60^(∘)-90^(∘) triangle whose hypotenuse is s. To find the apothem, we can use the trigonometric ratios.
cos( 30^(∘)) = a/s
⇒
a = s*cos( 30^(∘))
By using a calculator we get that cos( 30^(∘)) = sqrt(3)/2. Finally, we find the area of the hexagon by halving the product between the apothem and the perimeter.
ccl
A_H = 1/2* a* P
& ⇒ &
A_H = 1/2* ssqrt(3)/2 * 6 s [0.3cm]
& & A_H = 3 s^2sqrt(3)/2
