Tess builds a sandbox on the shape of a regular hexagon in which all sides are 40 inches long
To find the area of the sandbox, we need to calculate the area of the regular hexagon.
A regular hexagon can be divided into six congruent equilateral triangles. The area of an equilateral triangle can be calculated using the formula:
Area = (s^2 * √3) / 4,
where "s" is the length of each side.
In this case, the length of each side, s, is 40 inches.
Plugging in the values, we have:
Area of each equilateral triangle = (40^2 * √3) / 4
= (1600 * √3) / 4
= 400√3 square inches.
Since there are six equilateral triangles in a hexagon, the total area of the hexagon is:
Total area of hexagon = 6 * Area of each equilateral triangle
= 6 * 400√3 square inches
= 2400√3 square inches.
Thus, the area of Tess's sandbox, which is a regular hexagon with all sides measuring 40 inches, is 2400√3 square inches.
To find the area of the sandbox, we can use the formula for the area of a regular hexagon. The formula is:
Area = (3√3 * s^2) / 2
where "s" is the length of each side of the hexagon. In this case, the length of each side is 40 inches.
First, let's calculate the area using this formula:
Area = (3√3 * 40^2) / 2
To simplify the calculation, we can solve it step by step:
Step 1: Calculate the square of 40: 40 * 40 = 1600
Step 2: Multiply the square by 3√3: 1600 * 3√3
Step 3: Divide the result by 2: (1600 * 3√3) / 2
Step 4: Multiply the value inside the square root (√3) by 3: (1600 * 3 * 3√3) / 2
Step 5: The final calculation is: (1600 * 9√3) / 2
To find the actual value, we need to find the approximate value of √3.
Step 6: Refer to a calculator or calculate the approximated value of √3, which is approximately 1.732.
Step 7: Substitute the value of √3 in the equation: (1600 * 9 * 1.732) / 2
Step 8: Simplify the equation: (13824 * 1.732) / 2
Step 9: Calculate the final result: 23835.264 / 2 = 11917.632
Therefore, the area of the sandbox is approximately 11,917.632 square inches.
To find the area of a regular hexagon, you need to know the length of one side. In this case, all the sides of the hexagon are 40 inches long.
Step 1: Find the apothem (distance from the center of the hexagon to a side).
The apothem of a regular hexagon can be calculated using the formula a = (s√3)/2, where s is the length of a side.
a = (40√3)/2
a = 20√3
Step 2: Find the area of the hexagon using the formula A = (nsa)/2, where n is the number of sides and a is the apothem.
A = (6 * 40 * 20√3)/2
A = (480√3) square inches
Therefore, the area of the sandbox is 480√3 square inches.