The area of a regular hexagon inscribed in a circle is equal to 166.28 square cm. If the circle is inscribed in a square, find the difference between the area of the square and the hexagon.
89.72 sq. cm
amogus
Why did the circle feel square? Because it was surrounded by hexagons! Let's solve this riddle and find the answer.
The area of a regular hexagon can be found using the formula: A = (3√3 × side²) / 2
Given that the area of the hexagon is 166.28 square cm, we can use this formula to find the length of one side.
166.28 = (3√3 × side²) / 2
By rearranging the equation, we find:
side² = (166.28 × 2) / (3√3)
Solving further, we get:
side² = 111.167
Taking the square root of both sides, we find:
side ≈ 10.54 cm (approximately)
Now, since the hexagon is inscribed in a circle, and the circle is inscribed in a square, the length of a side of the square is equal to the diameter of the circle.
The diameter of the circle = 2 × radius
The radius of the circle is equal to the distance from the center of the circle to one of the vertices of the hexagon.
Using the Pythagorean theorem, we can find the radius:
radius = √(side² + (side/2)²)
Plugging in the value of side, we get:
radius ≈ √(10.54² + (10.54/2)²)
radius ≈ √(111.167 + 27.835)
radius ≈ √139.002
radius ≈ 11.78 cm (approximately)
Now we can find the area of the square. The area of a square is given by A = side².
Area of the square = (11.78 cm)² = 138.3684 square cm
Finally, to find the difference between the area of the square and the hexagon:
Difference = Area of the square - Area of the hexagon
Difference ≈ 138.3684 - 166.28
Difference ≈ -27.9116
So, the difference between the area of the square and the hexagon is approximately -27.9116 square cm.
To find the difference between the area of the square and the hexagon, we first need to find the side length of the square.
Since the hexagon is inscribed in the circle, the distance from the center of the circle to any vertex of the hexagon equals the radius of the circle.
Now, the side length of the hexagon is twice the radius of the circle, since the radius forms two sides of the hexagon. Therefore, the side length of the hexagon is 2 * √(166.28 / 3) cm.
To find the diagonal of the square, we can use the side length of the hexagon. Since the diagonal of a square forms the hypotenuse of a right triangle with the side length of the square, the diagonal of the square is √2 times the side length of the square.
So, the diagonal of the square is √2 * 2 * √(166.28 / 3) cm.
The diagonal of the square forms two sides of an isosceles right triangle with the side length of the square as the base. Therefore, the side length of the square is (√2 * 2 * √(166.28 / 3)) / √2 cm.
Simplifying, we get the side length of the square as 2 * √(166.28 / 3) cm.
Now, the area of the hexagon is given by the formula (3√3 / 2) * s^2, where s is the side length of the hexagon.
So, the area of the hexagon is (3√3 / 2) * (2 * √(166.28 / 3))^2 square cm.
Simplifying further, we find the area of the hexagon is approximately 951.43 square cm.
Finally, to find the difference between the area of the square and the hexagon, we subtract the area of the hexagon from the area of the square.
Area of square - Area of hexagon = (2 * √(166.28 / 3))^2 - 951.43 square cm.
Now, we can calculate the difference in the areas of the square and the hexagon using the above formula.
To find the difference between the area of the square and the hexagon, we first need to find the side length of the square.
1. Find the radius of the circle:
The area of a regular hexagon inscribed in a circle is given by the formula: Area = (3√3 * s^2) / 2, where 's' is the side length of the hexagon.
Given that the area of the hexagon is 166.28 square cm, we can use this formula to find the side length of the hexagon.
(3√3 * s^2) / 2 = 166.28
Multiply both sides by 2 to get rid of the fraction: 3√3 * s^2 = 332.56
Divide both sides by 3√3: s^2 = 332.56 / (3√3)
Take the square root of both sides: s = √(332.56 / (3√3))
Calculate the value of s using a calculator: s ≈ 8.50 cm
2. Find the diameter of the circle:
The diameter of the circle is equal to twice the radius. So, multiply the side length of the hexagon by 2 to find the diameter.
Diameter = 2 * (√(332.56 / (3√3)))
Calculate the value of the diameter using a calculator.
3. Find the side length of the square:
The diagonal of the square is equal to the diameter of the circle. The diagonal of a square is √2 times its side length. So, divide the diameter by √2 to find the side length of the square.
Square side length = Diameter / √2
Calculate the value of the square side length using a calculator.
4. Find the area of the square:
The area of a square is given by the formula: Area = side length * side length.
Calculate the area of the square using the side length found in step 3.
5. Find the difference:
Subtract the area of the hexagon (166.28 square cm) from the area of the square to find the difference.
Difference = Area of square - Area of hexagon
By following these steps, you can calculate the difference between the area of the square and the hexagon.