The distance between two opposite vertices of the dodecagon is 2. Find the area of the dodecagon.
first, what is the area of a dodecagon and second how to find it with just the distance between two oppositve vertices.
i found that a dodecagon can be divided into 24 triangles: an apothem, half of one of the sides, and half of the distance between two oppositite vertices of the dodecagon. So how do we do this ?
BTW I did not learn trigonometry so this can be figured out through geometry.
Thanks a lot for helping me if you can.
the diagonal is the diameter of the circumscribed circle.
The area of a dodecagon inscribed in a circle of radius R is 3R^2.
So, here R=1, and A = 3
You can find a geometric derivation of the area in Euclid Book IV, but it's heavy going.
Hello,
Draw a diagram, splitting the dodecagon into 12 congruent isosceles triangles. The vertex angle of one of the congruent isosceles triangles is 30 degrees, so drop an altitude that isn't from the vertex angle's vertex. That forms a 30-60-90 triangle, and the altitude is 1/2. The area of one of those triangles is 1 * 1/2 * 1/2 = 1/4, and the area of the dodecagon is 12 * 1/4 = 3.
To find the area of a dodecagon, you can use the formula:
Area = (1/2) × Perimeter × Apothem
However, in order to use this formula, we need to know either the length of the side of the dodecagon or the apothem.
Since you have mentioned that you know the distance between two opposite vertices of the dodecagon, we can use this information to find the apothem.
To do this, let's draw a line segment connecting two opposite vertices of the dodecagon. This line segment passes through the center of the dodecagon, dividing it into two congruent triangles.
Next, draw a line segment from the center of the dodecagon to one of the endpoints of the line segment connecting opposite vertices. This line segment is called the apothem.
Now, we have formed a right-angled triangle with the apothem as one of the legs and the distance between opposite vertices as the hypotenuse.
Since a dodecagon has 12 sides, the distance between two opposite vertices is the same as the perimeter of the dodecagon.
Therefore, we can consider the right-angled triangle formed as follows:
Let's assume that the distance between the opposite vertices is 2. So, the perimeter of the dodecagon is 2.
Using the Pythagorean theorem, we can find the length of the apothem.
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In our case, the hypotenuse is the distance between opposite vertices (2) and the apothem is the other shorter side. Let's denote the apothem as 'a' for simplicity.
So, we have the equation:
a^2 + a^2 = 2^2
Simplifying this equation, we get:
2a^2 = 4
a^2 = 2
Taking the square root of both sides, we find:
a = √2
Now that we have the length of the apothem (√2), we can find the area using the formula mentioned earlier:
Area = (1/2) × Perimeter × Apothem
Plugging in the values:
Area = (1/2) × 2 × √2
Area = √2
Therefore, the area of the dodecagon is √2 square units.