Find the area of a circle circumscribed about a regular hexagon with an apothem of 5 inches. Round to the nearest tenth
To find the area of the circle circumscribed about a regular hexagon, we need to determine the radius of the circle first.
The apothem of a regular hexagon is the distance from the center to any side. Given that the apothem is 5 inches, we can find the radius using the following formula:
Radius (r) = apothem (a) / cosine(30°)
Since the regular hexagon has angles of 120° at each vertex, we can divide it into six congruent equilateral triangles with angles of 60°.
The cosine of 30° is equal to (√3)/2. Substituting this value, we have:
Radius (r) = 5 / (√3)/2
To simplify this expression, we multiply the fraction by the reciprocal of (√3)/2:
Radius (r) = 5 * 2 / (√3)
= 10 / (√3)
= (10 * √3) / 3
Now that we have the radius, we can calculate the area of the circle using the formula:
Area (A) = π * r²
Substituting in the value of the radius:
Area (A) = π * [(10 * √3) / 3]²
To round to the nearest tenth, we can use the approximation:
π ≈ 3.14
Area (A) ≈ 3.14 * [(10 * √3) / 3]²
≈ 3.14 * (100 * 3) / 9
≈ 3.14 * 33.33
≈ 104.67
Therefore, the area of the circle, rounded to the nearest tenth, is approximately 104.7 square inches.
To find the area of a circle circumscribed about a regular hexagon, we can use the formula:
Area of the circle = (3√3/2) * (apothem^2)
Given that the apothem of the regular hexagon is 5 inches, we can substitute this value into the formula:
Area of the circle = (3√3/2) * (5^2)
First, calculate the square of the apothem:
Apothem^2 = 5^2 = 25
Next, calculate √3:
√3 ≈ 1.732
Now, substitute the values into the formula:
Area of the circle = (3 * 1.732/2) * 25
Simplify:
Area of the circle ≈ (5.196) * 25
Area of the circle ≈ 129.9
Rounding to the nearest tenth, the area of the circle circumscribed about the regular hexagon is approximately 129.9 square inches.
I am not certain of your comfort with this, however, knowing either the apotherm or the circumradius you can find the other.
circumradius * cos(180/n)=apotherm
so in this case, atotherm=5inches, n=6
so circumradius * cos(30)=5inches
radius= 5inches / .866
Now the area of the circumscribed circle is PI*radius^2
Neat stuff. Thank your teacher.
http://en.wikipedia.org/wiki/Apothem