Squares - Math
Card 1 of 136
Freddie is building a square pen for his pig. He plans to buy x feet of fencing to build the pen. This will result in a pen with an area of p square feet. Unfortunately, he only has enough money to buy one third of the planned amount of fencing. Which expression represents the area of the pen he can build with this limited amount of fencing?
Freddie is building a square pen for his pig. He plans to buy x feet of fencing to build the pen. This will result in a pen with an area of p square feet. Unfortunately, he only has enough money to buy one third of the planned amount of fencing. Which expression represents the area of the pen he can build with this limited amount of fencing?
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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.
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.
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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.
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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?
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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
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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?
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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?
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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?
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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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If the diagonal of a square measures 
, what is the area of the square?
If the diagonal of a square measures
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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 
This is an isosceles right triangle, so the diagonal must equal
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If the perimeter of a square is 44 centimeters, what is the area of the square in square centimeters?
If the perimeter of a square is 44 centimeters, what is the area of the square in square centimeters?
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Since the square's perimeter is 44, then each side is 
.
Then in order to find the area, use the definition that the
Since the square's perimeter is 44, then each side is
Then in order to find the area, use the definition that the
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A square 
has side lengths of 
. A second square 
has side lengths of 
. How many 
can you fit in a single 
?
A square
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The area of 
is 
, the area of 
is 
. Therefore, you can fit 5.06 
in 
.
The area of
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The perimeter of a square is 
If the square is enlarged by a factor of three, what is the new area?
The perimeter of a square is
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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 
.
The perimeter of a square is given by
So the area of the new square is given by
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Given square 
, with midpoints on each side connected to form a new, smaller square. How many times bigger is the area of the larger square than the smaller square?
Given square
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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.
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
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.
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Eric has 160 feet of fence for a parking lot he manages. If he is using all of the fencing, what is the area of the lot assuming it is square?
Eric has 160 feet of fence for a parking lot he manages. If he is using all of the fencing, what is the area of the lot assuming it is square?
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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.
.
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.
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Find the area of a square with a diagonal of 
.
Find the area of a square with a diagonal of
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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 
.
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
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If the ratio of the sides of two squares is 
, what is the ratio of the areas of those two squares?
If the ratio of the sides of two squares is
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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 
.
Express the ratio of the two sides of the squares as
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What is the area of a square with a diagonal of 
?
What is the area of a square with a diagonal of
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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:
The formula for the area of a square is
Remember that all sides of a square are equal, so the diagonal cuts the square into two equal triangles, each a
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:
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The perimeter of a square is 48. What is the length of its diagonal?
The perimeter of a square is 48. What is the length of its diagonal?
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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:
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:
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What is the length of a diagonal of a square with a side length 
? Round to the nearest tenth.
What is the length of a diagonal of a square with a side length
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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.
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
In this instance,
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What is the length of the diagonal of a 7-by-7 square? (Round to the nearest tenth.)
What is the length of the diagonal of a 7-by-7 square? (Round to the nearest tenth.)
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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 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
Plug the side lengths into the equation as
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.
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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
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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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