You are planning to use a ceramic tile design in your new bathroom. The tiles are blue-and-white equilateral triangles. You decide to arrange the blue tiles in a hexagonal shape as shown. If the side of each tile measures 7 centimeters, what will be the exact area of each hexagonal shape?
These answers are both wrong. It is 73.5^3cm2
73.5^3cm2
It takes 6 equilateral triangles to make a hexagon. Each equilaterial triangle area is
A = (1/2)(base)*(height)
= (1/2)(7)*(7 cos30)
= 34.2 cm^2
Multiply that by 6
I don't get the significance of the blue and white pattern.
Well, I guess you could say you're really "tri"-ing to tile your bathroom with humor! Anyway, to find the area of a hexagon, we can break it down into six equilateral triangles. Now, since each side of the blue tile measures 7 centimeters, we know that the height of each triangle is 7 centimeters.
To find the base of each equilateral triangle, we can use a little geometry magic. The formula for the base, given the side length (which in this case is also 7 centimeters), is b = s √3, where b is the base length and s is the side length.
So, substituting in our values: b = 7 √3. Now we can calculate the area of each equilateral triangle using the formula: A = (base × height) / 2.
Plugging in our values: A = (7 √3 × 7) / 2. Simplifying this, we get A = 49√3 / 2.
Finally, since we have six identical equilateral triangles in our hexagon, we can find the total area by multiplying the area of one triangle by 6: 6 × (49√3 / 2).
Doing the math: 6 × 49√3 / 2 = 294√3 / 2.
And there you have it! The exact area of each hexagonal shape formed by the blue tiles will be 294√3 / 2 square centimeters.
To find the area of the hexagonal shape, we first need to find the area of one equilateral triangle.
The formula to find the area of an equilateral triangle is given by:
Area = (√3/4) * side^2
Given that the side of each tile measures 7 centimeters, we can substitute this value into the formula and calculate the area of one triangle.
Area of one triangle = (√3/4) * (7 cm)^2
Now, to find the area of the hexagonal shape, we need to consider that a hexagon is made up of six equilateral triangles. Since each blue tile represents one equilateral triangle, the area of the hexagonal shape will be six times the area of one triangle.
Area of hexagonal shape = 6 * Area of one triangle
= 6 * [ (√3/4) * (7 cm)^2 ]
Now we can calculate the area:
Area of hexagonal shape = 6 * [ (√3/4) * (7 cm)^2 ]
= 6 * [ (√3/4) * 49 cm^2 ]
= 6 * (√3/4) * 49 cm^2
= 3√3 * 49 cm^2
= 147√3 cm^2
Therefore, the exact area of each hexagonal shape is 147√3 square centimeters.