A regular hexagon with a perimeter of 24 units is inscribed in a circle. Find the radius of the circle.
Why did the hexagon go to the circus? Because it wanted to be inscribed in a "circle-us"! Now, let's tackle your question, shall we?
A regular hexagon has 6 equal sides, so each side of this hexagon would measure 24 divided by 6, which is 4 units.
The radius of the circle would be equal to the distance from the center of the circle to any of its vertices. To find that distance, we can consider a right triangle formed by the radius, half of the side length, and the distance from the center of the hexagon to the midpoint of any side.
Applying the Pythagorean theorem, we have:
(radius)^2 = (half of side length)^2 + (distance to the midpoint)^2
(radius)^2 = 4^2 + (radius - 4)^2
Expanding and simplifying:
(radius)^2 = 16 + (radius^2 - 8r + 16)
Combining like terms:
0 = 32 - 8r
Rearranging:
8r = 32
Dividing both sides by 8:
r = 4
So, the radius of the circle is 4 units. Ta-da!
To find the radius of the circle inscribed in a regular hexagon, we need to follow these steps:
1. Determine the length of each side of the hexagon.
2. Calculate the circumference of the circle using the perimeter of the hexagon.
3. Divide the circumference by 2π to find the radius.
Let's go step-by-step:
1. Determine the length of each side of the hexagon:
Since the hexagon is regular, all its sides are equal in length. So, we can find the length of one side by dividing the perimeter by 6 (the number of sides):
Perimeter of the hexagon = 24 units
Length of one side = Perimeter / Number of sides = 24 / 6 = 4 units
2. Calculate the circumference of the circle:
The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle.
In this case, the circumference of the circle is equal to the perimeter of the hexagon, which is 24 units.
Therefore, 2πr = 24
3. Divide the circumference by 2π to find the radius:
Dividing both sides of the equation by 2π, we get:
r = 24 / (2π) ≈ 3.82 units
Hence, the radius of the inscribed circle in the regular hexagon is approximately 3.82 units.
To find the radius of the circle inscribed in a regular hexagon, we can follow these steps:
Step 1: Determine the length of one side of the hexagon.
Since a regular hexagon has six equal sides, we can divide the perimeter (24) by 6 to find the length of one side.
Side length = Perimeter / Number of Sides = 24 / 6 = 4 units
Step 2: Find the apothem of the hexagon.
The apothem is the distance from the center of the hexagon to the midpoint of one side. For a regular hexagon, it is also the radius of the inscribed circle.
Apothem = (Side Length / 2) × (√3) = (4 / 2) × (√3) = 2√3 units
Therefore, the radius of the inscribed circle is equal to the apothem of the hexagon, which is 2√3 units.
A hexagon consists of 6 equilateral triangles.
Therefore the side of each triangle is 24/6=4 units which also equals the radius of the circumscribed circle.