A regular polygon is inscribed in a circle of radius 11 cm. Calculate the perimeter of the polygon to the nearest tenth if the polygon has 8 sides.
Why did the polygon join the circus? Because it wanted to be a well-rounded performer!
Now, let's calculate the perimeter of the polygon. In an inscribed regular polygon, each side of the polygon is a chord of the circle. To find the perimeter, we need to determine the length of one side of the polygon.
To do this, we form a right triangle inside the polygon by connecting the center of the circle to two consecutive vertices. The hypotenuse of this right triangle is the radius of the circle, which is 11 cm.
Since the polygon has 8 sides, the central angle (angle formed at the center of the circle) between each consecutive pair of vertices is 360°/8 = 45°.
Using the trigonometric relationship "cosine," we can find the length of one side (x) of the polygon:
cos(45°) = adjacent/hypotenuse
cos(45°) = x/11
x = 11 * cos(45°) ≈ 7.8 cm
Since the polygon has 8 sides, the perimeter is 8 * 7.8 cm ≈ 62.4 cm.
So, the approximate perimeter of the polygon is 62.4 cm.
To calculate the perimeter of a regular polygon inscribed in a circle, you need to know the radius of the circle and the number of sides of the polygon. In this case, the radius of the circle is given to be 11 cm and the polygon has 8 sides.
Step 1: Find the central angle of the polygon.
The central angle of a regular polygon can be calculated using the formula:
central angle = 360° / number of sides
central angle = 360° / 8 = 45°
Step 2: Calculate the length of one side of the polygon.
The central angle of a regular polygon creates an isosceles triangle in which the side opposite the central angle is the same as the radius of the circle.
Using the given radius of 11 cm, we can find the length of one side by using the trigonometric relationship:
side length = 2 × radius × sin(central angle/2)
side length = 2 × 11 cm × sin(45°/2) ≈ 2 × 11 cm × 0.383 = 8.427 cm
Step 3: Calculate the perimeter of the polygon.
Since the regular polygon has 8 sides, the perimeter can be calculated by multiplying the length of one side by the number of sides:
perimeter = side length × number of sides
perimeter ≈ 8.427 cm × 8 = 67.416 cm
Rounding the perimeter to the nearest tenth, the final answer is approximately 67.4 cm.
To calculate the perimeter of a regular polygon, we need to find the length of one side and then multiply it by the number of sides.
In this case, since the regular polygon is inscribed in a circle, the distance from the center of the circle to any vertex (the radius) forms a right triangle with half of one side as the hypotenuse.
Let's call the length of one side of the polygon "s". The radius of the circle is given as 11 cm, which is equal to half the length of one side plus the apothem (the distance from the center of the polygon to the midpoint of one side).
Using the Pythagorean theorem, we can find the length of one side:
s/2 = 11* sin(π/8)
Now, let's calculate this value: