find the area of a regular pentagram if one of the sides of inner pentagon is 3cm
50.1 sq.cm.
http://www.contracosta.edu/legacycontent/math/stararea.htm
To find the area of a regular pentagram, we need to know the side length of the inner pentagon or the radius of the circumcircle.
In this case, you mentioned that one side of the inner pentagon is 3 cm. However, to calculate the area of a regular pentagram, we need the side length of the entire pentagram. Let's assume that the given 3 cm refers to the side length of the entire pentagram.
To find the area of a regular pentagram, we need to split it into three different shapes: an inner pentagon, five isosceles triangles, and five congruent triangles.
First, let's find the area of the inner pentagon:
The formula for the area of a regular polygon is A = (1/4) * n * s^2 * cot(π/n), where:
- A is the area of the polygon
- n is the number of sides in the polygon
- s is the length of each side
In this case, the inner pentagon has 5 sides (n=5) and a side length of 3 cm (s=3). Plugging these values into the formula, we get:
A_inner_pentagon = (1/4) * 5 * 3^2 * cot(π/5)
Next, let's find the area of one isosceles triangle:
To calculate the area of an isosceles triangle, we need its base and height. In a regular pentagram, each isosceles triangle has two equal sides and one base that is a side of the inner pentagon. Since we already know the side length of the inner pentagon (3 cm), it serves as the base for each isosceles triangle.
To find the height of the isosceles triangle, we can use the Pythagorean theorem. Bisecting the isosceles triangle will give us a right triangle. Since we know the base (3 cm) and one of the equal sides (half the side of the inner pentagon), we can calculate the height of the isosceles triangle.
Now, let's denote the length of half the side of the inner pentagon as a. According to the Pythagorean theorem, we have:
a^2 + h^2 = (3/2)^2
h^2 = (3/2)^2 - a^2
h^2 = 9/4 - a^2
h = sqrt(9/4 - a^2)
Once we have the height, the formula to find the area of one isosceles triangle is:
A_isosceles_triangle = (1/2) * base * height
A_isosceles_triangle = (1/2) * 3 * sqrt(9/4 - a^2)
Finally, to get the total area of the pentagram, we multiply the area of one isosceles triangle by 5 (since there are five of them), and add the area of the inner pentagon:
A_pentagram = 5 * A_isosceles_triangle + A_inner_pentagon
Plug in the value of a (half the side of the inner pentagon) into the equation to find the area of the pentagram.