In a circle with a diameter of 10m, a regular five-pointed star touching its circumference is inscribed. What is the area of the part not covered by the star? Pls help with a solution and a diagram. Thanks!
google is your friend. You might start here:
http://www.contracosta.edu/legacycontent/math/stararea.htm
50.48 m^2
To find the area not covered by the star, we first need to calculate the area of the circle and the area of the star.
1. Calculate the Area of the Circle:
The formula for the area of a circle is A = πr^2, where A is the area and r is the radius. We are given the diameter, which is equal to twice the radius. Therefore, the radius (r) of the circle is 10m / 2 = 5m.
Using the formula, the area of the circle = π * (5m)^2 = 25π square meters.
2. Calculate the Area of the Star:
To calculate the area of the star, we first need to find the length of one of its sides. In a regular five-pointed star, each interior angle is 36 degrees.
Since the sum of all the interior angles of a pentagon is 540 degrees, and the star consists of 5 congruent triangles, each triangle's interior angle will be 540 degrees / 5 = 108 degrees.
To find the length of one side of the star, we can use the Law of Cosines:
c^2 = a^2 + b^2 - 2ab * cos(C),
where a and b are the lengths of the triangle's sides and C is the angle opposite side c. Since we want to find the length of one side, both a and b are the same.
Using the Law of Cosines, we can find the length of one side of the triangle:
c^2 = 10^2 + 10^2 - 2 * 10 * 10 * cos(108) = 400 - 200 * cos(108).
Using a calculator, we get c ≈ 7.88 meters.
Now that we know the length of one side of the triangle, we can calculate the area of the star by using the formula for the area of a regular pentagon:
A = (5/4) * c^2 * tan(180/5).
Using the values, the area of the star ≈ (5/4) * (7.88m)^2 * tan(36) ≈ 38.04 square meters.
3. Calculate the Area Not Covered by the Star:
To find the area not covered by the star, we subtract the area of the star from the area of the circle:
Area not covered = Area of the Circle - Area of the Star
= 25π - 38.04
≈ 25π - 38.04 ≈ 64.66 square meters.
Therefore, the area not covered by the star is approximately 64.66 square meters.
[Diagram]
(The star is not drawn to scale)
```
_____
/¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\
/ \
\
\______ ______/
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```
To find the area of the part not covered by the star, we first need to find the area of the entire circle and the area of the star.
1. Area of the Circle:
The formula for 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 10m, we can find the radius by dividing the diameter by 2.
radius = diameter / 2 = 10m / 2 = 5m
Using the formula, we can calculate the area of the circle:
A_circle = π × (5m)²
A_circle = 25π m²
2. Area of the Star:
To find the area of the star, we can divide it into five triangles.
Each triangle can be considered an isosceles triangle with two equal sides and an angle of 36 degrees at the center of the circle.
The base of each triangle is the radius of the circle (5m), and the height can be found using trigonometry.
The formula to find the height (h) of an isosceles triangle is h = b × sin(angle), where b is the base and angle is the central angle.
h = 5m × sin(36°)
h ≈ 5m × 0.5878
h ≈ 2.939 m
The area of each triangle can be found using the formula A = (1/2) × base × height:
A_triangle = (1/2) × 5m × 2.939m
A_triangle ≈ 7.3475 m²
Since there are five identical triangles, the total area of the star is:
A_star = 5 × A_triangle
A_star ≈ 5 × 7.3475 m²
A_star ≈ 36.7375 m²
3. Area of the Part Not Covered by the Star:
The area of the part not covered by the star is equal to the area of the circle minus the area of the star.
A_not_covered = A_circle - A_star
A_not_covered = 25π m² - 36.7375 m²
A_not_covered ≈ (25π - 36.7375) m²
Using a calculator, you can evaluate the approximate value of A_not_covered.
Unfortunately, I cannot generate a diagram here, but you can visualize it by drawing a circle with a diameter of 10m and inscribing a regular five-pointed star within it. The area outside the star represents the part not covered.