Squares - Math
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What is the length of the diagonal of a square with a side length of 
?
What is the length of the diagonal of a square with a side length of
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 
.
To find the diagonal of a square, we must use the side length to create a 90 degree triangle with side lengths of
Pythagorean’s Theorem states
Take
After squaring the numbers, add them together:
Once you have the sum, take the square root of both sides:
Simplify to find the answer:
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A square has sides of 
. What is the length of the diagonal of this square?
A square has sides of
To find the diagonal of the square, we effectively cut the square into two 
triangles.
The pattern for the sides of a 
is 
.
Since two sides are equal to 
, this triangle will have sides of 
.
Therefore, the diagonal (the hypotenuse) will have a length of 
.
To find the diagonal of the square, we effectively cut the square into two
The pattern for the sides of a
Since two sides are equal to
Therefore, the diagonal (the hypotenuse) will have a length of
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A square has sides of 
. What is the length of the diagonal of this square?
A square has sides of
To find the diagonal of the square, we effectively cut the square into two 
triangles.
The pattern for the sides of a 
is 
.
Since two sides are equal to 
, this triangle will have sides of 
.
Therefore, the diagonal (the hypotenuse) will have a length of 
.
To find the diagonal of the square, we effectively cut the square into two
The pattern for the sides of a
Since two sides are equal to
Therefore, the diagonal (the hypotenuse) will have a length of
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The area of square R is 12 times the area of square T. If the area of square R is 48, what is the length of one side of square T?
The area of square R is 12 times the area of square T. If the area of square R is 48, what is the length of one side of square T?
We start by dividing the area of square R (48) by 12, to come up with the area of square T, 4. Then take the square root of the area to get the length of one side, giving us 2.
We start by dividing the area of square R (48) by 12, to come up with the area of square T, 4. Then take the square root of the area to get the length of one side, giving us 2.
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When the side of a certain square is increased by 2 inches, the area of the resulting square is 64 sq. inches greater than the original square. What is the length of the side of the original square, in inches?
When the side of a certain square is increased by 2 inches, the area of the resulting square is 64 sq. inches greater than the original square. What is the length of the side of the original square, in inches?
Let x represent the length of the original square in inches. Thus the area of the original square is x2. Two inches are added to x, which is represented by x+2. The area of the resulting square is (x+2)2. We are given that the new square is 64 sq. inches greater than the original. Therefore we can write the algebraic expression:
x2 + 64 = (x+2)2
FOIL the right side of the equation.
x2 + 64 = x2 + 4x + 4
Subtract x2 from both sides and then continue with the alegbra.
64 = 4x + 4
64 = 4(x + 1)
16 = x + 1
15 = x
Therefore, the length of the original square is 15 inches.
If you plug in the answer choices, you would need to add 2 inches to the value of the answer choice and then take the difference of two squares. The choice with 15 would be correct because 172 -152 = 64.
Let x represent the length of the original square in inches. Thus the area of the original square is x2. Two inches are added to x, which is represented by x+2. The area of the resulting square is (x+2)2. We are given that the new square is 64 sq. inches greater than the original. Therefore we can write the algebraic expression:
x2 + 64 = (x+2)2
FOIL the right side of the equation.
x2 + 64 = x2 + 4x + 4
Subtract x2 from both sides and then continue with the alegbra.
64 = 4x + 4
64 = 4(x + 1)
16 = x + 1
15 = x
Therefore, the length of the original square is 15 inches.
If you plug in the answer choices, you would need to add 2 inches to the value of the answer choice and then take the difference of two squares. The choice with 15 would be correct because 172 -152 = 64.
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If the area of a square is 25 inches squared, what is the perimeter?
If the area of a square is 25 inches squared, what is the perimeter?
The area of a square is equal to length times width or length squared (since length and width are equal on a square). Therefore, the length of one side is 
or 
The perimeter of a square is the sum of the length of all 4 sides or 
The area of a square is equal to length times width or length squared (since length and width are equal on a square). Therefore, the length of one side is
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A square is inscribed inside a circle, as illustrated above. The radius of the circle is 
units. If all of the square's diagonals pass through the circle's center, what is the area of the square?
A square is inscribed inside a circle, as illustrated above. The radius of the circle is
Given that the square's diagonals pass through the circle's center, those diagonals must each form a diameter of the circle. The circle's diameter is twice its radius, i.e. 
, which is 
. Since this diameter (i.e., the square's diagonal) is the hypotenuse of a right triangle formed by two sides of the square, the length of one of the square's sides can be calculated with the Pythagorean Theorem. 
replace 
and 
because the sides of the square must be equal in length. Since the objective is to solve for the square's area, solve for 
since one side squared will be the square's area.
units squared
Given that the square's diagonals pass through the circle's center, those diagonals must each form a diameter of the circle. The circle's diameter is twice its radius, i.e.
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The diagonal of a square has a length of 10 inches. What is the perimeter of the square in inches squared?
The diagonal of a square has a length of 10 inches. What is the perimeter of the square in inches squared?
Using the Pythagorean Theorem, we can find the edge of a side to be √50, by 2a2=102. This can be reduced to 5√2. This can then be multiplied by 4 to find the perimeter.
Using the Pythagorean Theorem, we can find the edge of a side to be √50, by 2a2=102. This can be reduced to 5√2. This can then be multiplied by 4 to find the perimeter.
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A circle with a radius 2 in is inscribed in a square. What is the perimeter of the square?
A circle with a radius 2 in is inscribed in a square. What is the perimeter of the square?
To inscribe means to draw inside a figure so as to touch in as many places as possible without overlapping. The circle is inside the square such that the diameter of the circle is the same as the side of the square, so the side is actually 4 in. The perimeter of the square = 4s = 4 * 4 = 16 in.
To inscribe means to draw inside a figure so as to touch in as many places as possible without overlapping. The circle is inside the square such that the diameter of the circle is the same as the side of the square, so the side is actually 4 in. The perimeter of the square = 4s = 4 * 4 = 16 in.
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Square X has 3 times the area of Square Y. If the perimeter of Square Y is 24 ft, what is the area of Square X, in sq ft?
Square X has 3 times the area of Square Y. If the perimeter of Square Y is 24 ft, what is the area of Square X, in sq ft?
Find the area of Square Y, then calculate the area of Square X.
If the perimeter of Square Y is 24, then each side is 24/4, or 6.
A = 6 * 6 = 36 sq ft, for Square Y
If Square X has 3 times the area, then 3 * 36 = 108 sq ft.
Find the area of Square Y, then calculate the area of Square X.
If the perimeter of Square Y is 24, then each side is 24/4, or 6.
A = 6 * 6 = 36 sq ft, for Square Y
If Square X has 3 times the area, then 3 * 36 = 108 sq ft.
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A square garden has an area of 64 square feet. If you add 3 feet to each side, what is the new perimeter of the garden?
A square garden has an area of 64 square feet. If you add 3 feet to each side, what is the new perimeter of the garden?
By finding the square root of the area of the garden, you find the length of one side, which is 8. We add 3 feet to this, giving us 11, then multiply this by 4 to get 44 feet for the perimeter.
By finding the square root of the area of the garden, you find the length of one side, which is 8. We add 3 feet to this, giving us 11, then multiply this by 4 to get 44 feet for the perimeter.
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The area of a square is 
. If the square is enlarged by a factor of 2, what is the perimeter of the new square?
The area of a square is
The area of a square is given by 
so we know the side is 5 cm. Enlarging by a factor of two makes the new side 10 cm. The perimeter is given by 
, so the perimeter of the new square is 40 cm.
The area of a square is given by
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A square has an area of 
. If the side of the square is reduced by a factor of two, what is the perimeter of the new square?
A square has an area of
The area of the given square is given by 
so the side must be 6 in. The side is reduced by a factor of two, so the new side is 3 in. The perimeter of the new square is given by 
.
The area of the given square is given by
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The area of the shaded region of a square is 18. What is the perimeter of the square?
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The area of the shaded region of a square is 18. What is the perimeter of the square?
The area of the shaded region, which covers half of the square is 18 meaning that the total area of the square is 18 x 2, or 36. The area of a square is equal to the length of one side squared. Since the square root of 36 is 6, the length of 1 side is 6. The perimeter is the length of 1 side times 4 or 6 x 4.
The area of the shaded region, which covers half of the square is 18 meaning that the total area of the square is 18 x 2, or 36. The area of a square is equal to the length of one side squared. Since the square root of 36 is 6, the length of 1 side is 6. The perimeter is the length of 1 side times 4 or 6 x 4.
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How much more area does a square with a side of 2r have than a circle with a radius r? Approximate π by using 22/7.
How much more area does a square with a side of 2r have than a circle with a radius r? Approximate π by using 22/7.
The area of a circle is given by A = πr2 or 22/7r2
The area of a square is given by A = s2 or (2r)2 = 4r2
Then subtract the area of the circle from the area of the square and get 6/7 square units.
The area of a circle is given by A = πr2 or 22/7r2
The area of a square is given by A = s2 or (2r)2 = 4r2
Then subtract the area of the circle from the area of the square and get 6/7 square units.
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ABCD and EFGH are squares such that the perimeter of ABCD is 3 times that of EFGH. If the area of EFGH is 25, what is the area of ABCD?
ABCD and EFGH are squares such that the perimeter of ABCD is 3 times that of EFGH. If the area of EFGH is 25, what is the area of ABCD?
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.
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.
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A half circle has an area of 
. What is the area of a square with sides that measure the same length as the diameter of the half circle?
A half circle has an area of
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.
If the area of the half circle is
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.
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If a completely fenced-in square-shaped yard requires 140 feet of fence, what is the area, in square feet, of the lot?
If a completely fenced-in square-shaped yard requires 140 feet of fence, what is the area, in square feet, of the lot?
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.
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.
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A square has an area of 36. If all sides are doubled in value, what is the new area?
A square has an area of 36. If all sides are doubled in value, what is the new area?
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.
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.
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If the perimeter of a square is equal to twice its area, what is the length of one of its sides?
If the perimeter of a square is equal to twice its area, what is the length of one of its sides?
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
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
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