What is the area of a square with a diagonal length of 15 times the square root of 2?
The diagonal of a square is the side times sqrt 2 (Pythagorean theorem)
so
the side here is 15 so the area is 15^2 = 225
To find the area of a square with a diagonal length of 15 times the square root of 2, we can follow these steps:
Step 1: Find the length of one side of the square.
The diagonal of a square forms a right triangle, with the side length as the hypotenuse. Using the Pythagorean theorem, we can find the length of one side:
a² + b² = c², where a and b are the legs (the sides of the right triangle) and c is the hypotenuse.
In this case, we know that c (the diagonal) is 15 times the square root of 2:
c = 15√2
Let's assume a and b are the sides of the square, so both a and b are equal. The equation becomes:
a² + a² = (15√2)²
2a² = (15√2)²
2a² = 15² * (√2)²
2a² = 225 * 2
2a² = 450
Now we can solve for a:
a² = 450 / 2
a² = 225
a = √225
a = 15
So, the length of one side of the square is 15.
Step 2: Calculate the area of the square.
The area of a square is given by the formula: Area = side length × side length.
In this case, the side length of the square is 15, so the area can be calculated as follows:
Area = 15 × 15
Area = 225
Therefore, the area of the square with a diagonal length of 15 times the square root of 2 is 225 square units.
To find the area of a square, you need to know the length of one of its sides. In this case, we are given the length of the diagonal, which we can use to find the length of the sides.
Let's call the length of one side of the square "s".
In a square, the diagonal forms a right triangle with the sides of the square. The length of the diagonal is the hypotenuse of this right triangle, and the sides of the square are the legs.
Using the Pythagorean theorem, we can find the relationship between the diagonal and the sides of the square:
(diagonal)^2 = (side)^2 + (side)^2
(15√2)^2 = s^2 + s^2
450 = 2s^2
Divide both sides by 2:
225 = s^2
Now, take the square root of both sides to find the length of each side:
s = √225
s = 15
Therefore, each side of the square is 15 units long.
To find the area of the square, you multiply the length of one side by itself:
Area = s^2 = 15^2 = 225
So, the area of the square is 225 square units.