In this video we are going to find the area of regular hexagon and it's given, the whole perimeter is 18 cm
So, now here is the hexagon, now it's regular that means all the sides are equal
X, X and then X here. We want to find the area, so the area, because it's a regular hexagon, these triangles are congruent, they have got the same size
So if we can find the area of one triangle, we multiply it by 6 and we will have the area of the hexagon because we have got 6 triangles here
Now, let's concentrate on one triangle, now X, X is one side, so the perimeter is 6 times X and this is given as 18 cm
Now we divide the equation 6X equals 18 by 6 and that gives me one side is equal to 3 cm
So let's concentrate on one triangle, because the hexagon is regular, all these 6 triangles that make up the hexagon are congruent, they have got the same size
So if we can find the area of one, we just multiply it by 6 to have the total area of the hexagon
Now, this side is X equals 2, 3, we know that, now angle alpha, this angle alpha is equal to this angle, to this angle, to this angle and so on
So we have got 6 alpha equal to the angle around the point and that is 360 degrees
So alpha is 360 degrees divided by 6 and that gives me 60 degrees
So this angle here is 60 degrees. Now, because again the hexagon is regular, this side and this side are congruent, they are equal in length, and therefore this triangle is isosceles
And if it is isosceles, so let's call this angle beta and beta, so we have 2 beta plus alpha which is 60 degrees
So this beta plus this beta, 2 beta, is 60 degrees, that gives you the sum in triangle, 180 degrees
And therefore, 2 beta is equal to 180 minus 60, 120, which gives beta equal 60 degrees
So, let's go back to our triangle, this is 60 degrees and this is 60 degrees and the triangle is equilateral
Equilateral, so this is 3, this is 3 and this is 3
Now, how do you find the area? The reminder of a formula to find the area of triangle, let's make a better triangle perhaps
Let's suppose this is A and B, the length of these 2 sides, the angle between them let's say is omega
The area of this triangle is equal to 1 half sin the angle omega, this angle here, multiply A, multiply B
Now, let's go here and apply this formula to find the area of this triangle
1 half, A is 3, sin omega, omega is the angle between the 2, so sin 60, multiply by A is 3, B is also 3, and that gives me 1 half
The sin 60 square root of 3 over 2 and 3 times 3 gives me 9, hang on, yeah that gives me 9 here, so it's 9 over 4 square root of 3
Now, the total area of the hexagon, let's call it A, H, is 6 times the area of triangle, so it will be 6
multiply 9 over 4 square root of 3 and this simplifies, 6 divided by 4 that gives you 3 over 2, 27 over 2 square root of 3 centimetre square, this is the area of the hexagon
So, this is the area of our hexagon and this is the final result
So, in this video what we did, we found the area of hexagon without use of the main formula for the area of hexagon, there is a formula