in a circle with diameter of 12 inches, a regular five-pointed star is inscribed. what is the area of that part not covered by the star?
Not such an easy question.
Here is your problem with a really nice diagram.
http://www.contracosta.edu/legacycontent/math/stararea.htm
72. 7 sq inches
To find the area of the part not covered by the star in a circle with a diameter of 12 inches, we need to subtract the area of the star from the total area of the circle.
First, let's find the area of the circle:
The radius of the circle is half the diameter, so it is 12 / 2 = 6 inches.
The area of a circle is given by the formula: A = π * r^2, where A is the area and r is the radius.
Plugging in the values, A = π * 6^2 = 36π square inches.
Now, let's find the area of the star:
To determine the area of a regular five-pointed star, we'll divide it into smaller triangles.
The central angle of each triangle in the star is 360 degrees divided by the number of points, which is 5, so the central angle is 360 / 5 = 72 degrees.
The base of each triangle is the distance from the center of the circle to the point of the star, which is equal to the radius of the circle, so it is 6 inches.
To find the height of the triangle, we can use the half of the base as the adjacent side and the tangent of half the central angle as the opposite side.
The half of the base is 6 / 2 = 3 inches.
The tangent of half the central angle is tan(72/2) = tan(36) ≈ 0.7265.
So, the height of the triangle is 3 * 0.7265 ≈ 2.1795 inches.
The area of each triangle can be calculated using the formula: A = (base * height) / 2.
Plugging in the values, the area of each triangle is (6 * 2.1795) / 2 ≈ 6.5385 square inches.
Since there are five triangles in the star, the total area of the star is 5 * 6.5385 = 32.6925 square inches.
Finally, to find the area of the part not covered by the star, we subtract the area of the star from the total area of the circle:
Area not covered = Total area of the circle - Total area of the star
Area not covered = 36π - 32.6925
Area not covered ≈ 112.68 square inches.
To find the area of the part not covered by the star in a circle with a diameter of 12 inches, we need to subtract the area of the star from the area of the circle.
To start, let's find the area of the circle:
The formula to find the area of a circle is A = πr², where A is the area and r is the radius.
Given that the diameter of the circle is 12 inches, we can find the radius by dividing the diameter by 2:
r = 12 inches / 2 = 6 inches
Using the formula for the area of a circle:
A_circle = π(6 inches)² = 36π square inches
Next, we need to find the area of the star. A regular five-pointed star can be divided into five congruent triangles. The triangles can be calculated using trigonometry.
Using trigonometry, we can determine that each internal angle of the star is 36 degrees. Therefore, each angle of the triangle formed by the star is 36 degrees. Additionally, since the star is inscribed in the circle, the base of each triangle is a chord of the circle.
To find the area of one triangle, given that the radius of the circle is 6 inches, we use the formula for the area of a triangle:
A_triangle = (1/2) * base * height
The base of the triangle is one side of the pentagon formed by the star/intersection with the circle.
To find the height, we need to calculate the length of the perpendicular line that connects the center of the circle to one of the sides of the triangle.
Using trigonometry, we can calculate that the angle between the radius and that perpendicular line is 18 degrees (half of 36 degrees).
Therefore, using trigonometry, the height of the triangle is:
height = sin(18 degrees) * 6 inches = 0.3090 * 6 inches = 1.854 inches (rounded to three decimal places)
The base of the triangle is equal to the length of the chord, which we can calculate using trigonometry:
base = 2 * sin(18 degrees) * 6 inches = 0.6180 * 6 inches = 3.708 inches (rounded to three decimal places)
Now, we can find the area of one triangle:
A_triangle = (1/2) * 3.708 inches * 1.854 inches = 3.437 square inches (rounded to three decimal places)
Since the star is made up of five congruent triangles, the total area of the star is:
A_star = 5 * 3.437 square inches = 17.187 square inches (rounded to three decimal places)
Finally, we subtract the area of the star from the area of the circle to find the area of the part not covered by the star:
Area_not_covered = A_circle - A_star = 36π square inches - 17.187 square inches = (36π - 17.187) square inches ≈ 95.367 square inches (rounded to three decimal places)
Therefore, the approximate area of the part not covered by the star is 95.367 square inches.