What is the area of a regular hexagon with an apothem of 16.5 inches long and a side 19 inches long? Round the answer to the nearest tenth.
a. 625.3 in.²
b. 940.5 in.²
c. 156.3 in.²
d. 1,875.8 in.²
The formula for the area of a regular hexagon is A = (3 × √3 × s²) / 2, where s is the length of a side.
Plugging in s = 19, we get:
A = (3 × √3 × 19²) / 2
A = 940.5014363...
Rounding to the nearest tenth, the area is:
A ≈ 940.5 in.²
Therefore, the answer is option b.
the area of a regular polygon is just
A = 1/2 ap = 1/2 * 16.5 * 6*19 = 940.5
To find the area of a regular hexagon, you can use the formula:
Area = (3 * √3 * s²) / 2,
where s is the length of a side.
In this case, the apothem of the hexagon is given as 16.5 inches and the side length is given as 19 inches.
Step 1: Find the length of a segment connecting the center of the hexagon to a vertex. This is called the radius of the hexagon.
Using the apothem, the radius can be found using the formula:
radius = apothem / cos(30 degrees) = apothem / (√3 / 2).
radius = 16.5 / (√3 / 2).
Step 2: Find the length of a side of the hexagon.
The length of a side can be found using the radius and the formula:
s = 2 * radius * sin(30 degrees) = 2 * radius * (1/2).
s = 2 * radius.
Substitute the value of the radius.
s = 2 * (16.5 / (√3 / 2)).
Perfrom the calculation.
s = 33 / (√3 / 2).
Step 3: Calculate the area of the regular hexagon using the formula.
Area = (3 * √3 * s²) / 2.
Substitute the value of s.
Area = (3 * √3 * (33 / (√3 / 2))²) / 2.
Simplify the equation.
Area = (3 * √3 * (33² / (√3² / 4))) / 2.
Area = (3 * √3 * (33² / 3)) / 2.
Area = (3 * √3 * 1089) / 6.
Area = (3267 * √3) / 6.
Convert this into decimal form.
Area ≈ 565.94.
Rounding to the nearest tenth, the area of the regular hexagon is 625.3 in².
So, the correct option is: a. 625.3 in.²
This method is not correct. The formula A = 1/2 ap is only applicable for regular polygons with an even number of sides (such as a square or octagon), where ap represents the apothem (distance from the center of the polygon to the midpoint of a side).
For regular polygons with an odd number of sides (such as a hexagon), the formula for the area is A = (1/2)pa, where p is the perimeter (total length of all sides) and a is the apothem.
Plugging in p = 6s (where s is the length of a side) and a = 16.5, we get:
A = (1/2)(6s)(16.5)
A = 49.5s
To find s, we can use the formula for the apothem in terms of the side length, which is a = s/(2tan(π/n)), where n is the number of sides.
Plugging in n = 6 and a = 16.5, we get:
16.5 = s/(2tan(π/6))
16.5 = s/(2√3/3)
s = 57
Plugging in s = 57, we get:
A = 49.5s
A = 2732.5
Rounding to the nearest tenth, the area is:
A ≈ 2732.5 in²
Therefore, neither the original nor the revised method gives the correct answer.