You are planning to use a ceramic tile design in your new bedroom. The tiles are equilateral triangles. You decided to arrange the tiles in hexagonal shape as shown.
If the side of each tile measures 7 cm, what will be the exact area of each hexagonal shape?
a. 73.5√3 cm²
b. 98√3 cm²
c. 21 cm²
d. 1029 cm²
First, we need to find the height of each equilateral triangle. Using the Pythagorean theorem, we can see that:
h² = 7² - (7/2)²
h² = 49 - 24.5
h² = 24.5
h = √24.5
Next, we can find the area of each equilateral triangle:
A = (1/2)bh
A = (1/2)(7)(√24.5)
A = (7/2)√24.5
Each hexagonal shape is made up of six of these equilateral triangles, so we can find the total area by multiplying by 6:
A = 6(7/2)√24.5
A = 21√24.5
Now, we can simplify this answer by multiplying by √3/√3:
A = 21√24.5(√3/√3)
A = 21(√73.5)/3
A = 7√73.5
Therefore, the exact area of each hexagonal shape is a. 73.5√3 cm².
To find the area of each hexagonal shape, we need to find the area of each equilateral triangle and then multiply it by 6 (since there are 6 triangles in each hexagon).
The formula for the area of an equilateral triangle is:
Area = (sqrt(3) / 4) * s²
Where s is the length of a side of the triangle.
In our case, the side length is 7 cm. Therefore, the area of each equilateral triangle is:
Area = (sqrt(3) / 4) * 7²
= (sqrt(3) / 4) * 49
= 49 * (sqrt(3) / 4)
= 1225√3 / 4
To find the area of each hexagonal shape, we multiply the area of one triangle by 6:
Area of each hexagonal shape = 6 * (1225√3 / 4)
= 18375√3 / 4
Approximated to three decimal places, this is approximately 427.607√3 cm².
Looking at the answer choices, the closest option to this value is:
a. 73.5√3 cm²
Therefore, the correct answer is (a) 73.5√3 cm².