Squares - Math

Assign variables such that

One side of ABCD = a

and One side of EFGH = e

Note that all sides are the same in a square. Since the perimeter is the sum of all sides, according to the question:

4a = 3 x 4e = 12e or a = 3e

From that area of EFGH is 25,

e x e = 25 so e = 5

Substitute a = 3e so a = 15

We aren’t done. Since we were asked for the area of ABCD, this is a x a = 225.

If the area of the half circle is , then the area of a full circle is twice that, or .

Use the formula for the area of a circle to solve for the radius:

36π = πr2

r = 6

If the radius is 6, then the diameter is 12. We know that the sides of the square are the same length as the diameter, so each side has length 12.

Therefore the area of the square is 12 x 12 = 144.

Since the yard is square in shape, we can divide the perimeter(140ft) by 4, giving us 35ft for each side. We then square 35 to give us the area, 1225 feet.

Let S be the original side length. S*S would represent the original area. Doubling the side length would give you 2S*2S, simplifying to 4*(S*S), giving a new area of 4x the original, or 144.

Area of a square in terms of each of its sides:

Area = S x S

Perimeter of a square:

Perimeter = 4S

So if 'the perimeter of a square is equal to twice its area':

2 x Area = Perimeter

2 x \[S x S\] = \[4S\]; divide by 2:

S x S = 2S; divide by S:

S = 2

If Freddie uses x feet of fencing makes a square, each side must be x/4 feet long. The area of this square is (x/4)2 = _x_2/16 = p square feet.

If Freddie uses one third of x feet of fencing makes a square, each side must be x/12 feet long. The area of this square is (x/12)2 = _x_2/144 = 1/9(_x_2/16) = 1/9(p) = p/9 square feet.

Alternate method:

The scale factor between the small perimeter and the larger perimeter = 1 : 3. Since we're comparing area, a two-dimensional measurement, we can square the scale factor and see that the ratio of the areas is 12 : 32 = 1 : 9.

This is an isosceles right triangle, so the diagonal must equal times the length of a side. Thus, one side of the square measures , and the area is equal to

Since the square's perimeter is 44, then each side is .

Then in order to find the area, use the definition that the

The area of is , the area of is . Therefore, you can fit 5.06 in .

The perimeter of a square is given by so the side length of the original square is The side of the new square is enlarged by a factor of 3 to give

So the area of the new square is given by .

Assume that the length of each midpoint is 1. This means that the length of each side of the large square is 2, so the area of the larger square is 4 square units.

To find the area of the smaller square, first find the length of each side. Because the length of each midpoint is 1, each side of the smaller square is (use either the Pythagorean Theorem or notice that these right trianges are isoceles right trianges, so can be used).

The area then of the smaller square is 2 square units.

Comparing the area of the two squares, the larger square is 2 times larger than the smaller square.

The area of a square is equal to its length times its width, so we need to figure out how long each side of the parking lot is. Since a square has four sides we calculate each side by dividing its perimeter by four.

Each side of the square lot will use 40 feet of fence.

.

A few facts need to be known to solve this problem. Observe that the diagonal of the square cuts it into two right isosceles triangles; therefore, the length of a side of the square to its diagonal is the same as an isosceles right triangle's leg to its hypotenuse: .

Rearrange an solve for .

Now, solve for the area using the formula .

Express the ratio of the two sides of the squares as . The area of each square is one side multiplied by itself, so the ratios of the areas would be . The right side of this equation simplifies to a ratio of .

The formula for the area of a square is . However, the problem gives us a diagonal and not a side.

Remember that all sides of a square are equal, so the diagonal cuts the square into two equal triangles, each a right triangle.

If we use the Pythagorean Theorem, we see:

Plug in our given diagonal to solve.

From here we can plug our answer back into our original equation:

Perimeter = side * 4

48 = side * 4

Side = 12

We can break up the square into two equal right triangles. The diagonal of the sqaure is then the hypotenuse of these two triangles.

Therefore, we can use the Pythagorean Theorem to solve for the diagonal:

A square is comprised of two 45-45-90 right triangles. The hypotenuse of a 45-45-90 right triangle follows the rule below, where is the length of the sides.

In this instance, is equal to 6.

To find the diagonal of a square we must use the side lengths to create a 90 degree triangle with side lengths of 7 and a hypotenuse which is equal to the diagonal.

We can use the Pythagorean Theorem here to solve for the hypotenuse of a right triangle.

The Pythagorean Theorem states , where a and b are the sidelengths and c is the hypotenuse.

Plug the side lengths into the equation as and :

Square the numbers:

Add the terms on the left side of the equation together:

Take the square root of both sides:

Therefore the length of the diagonal is 9.9.

To find the diagonal of a square, we must use the side length to create a 90 degree triangle with side lengths of , , and a hypotenuse which is equal to the diagonal.

Pythagorean’s Theorem states , where a and b are the legs and c is the hypotenuse.

Take and and plug them into the equation for and :

After squaring the numbers, add them together:

Once you have the sum, take the square root of both sides:

Simplify to find the answer: , or .