Plane Geometry - Math
Card 1 of 2152
The length of a rectangle is 
and the width is 
. What is its perimeter?
The length of a rectangle is
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Perimeter of a quadrilateral is found by adding up the lengths of its sides. The formula for the perimeter of a rectangle is 
.
Perimeter of a quadrilateral is found by adding up the lengths of its sides. The formula for the perimeter of a rectangle is
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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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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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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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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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A regular polygon with 
sides has exterior angles that measure 
each. How many sides does the polygon have?
A regular polygon with
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The sum of the exterior angles of any polygon, one per vertex, is 
. As each angle measures 
, just divide 360 by 1.5 to get the number of angles.
The sum of the exterior angles of any polygon, one per vertex, is
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What is the interior angle measure of any regular nonagon?
What is the interior angle measure of any regular nonagon?
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To find the angle of any regular polygon you find the number of sides 
, which in this example is 
.
You then subtract 
from the number of sides yielding 
.
Take 
and multiply it by 
degrees to yield a total number of degrees in the regular nonagon.
Then to find one individual angle we divide 
by the total number of angles 
.
The answer is 
.
To find the angle of any regular polygon you find the number of sides
You then subtract
Take
Then to find one individual angle we divide
The answer is
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What is the measure of one exterior angle of a regular seventeen-sided polygon (nearest tenth of a degree)?
What is the measure of one exterior angle of a regular seventeen-sided polygon (nearest tenth of a degree)?
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The sum of the measures of the exterior angles of any polygon, one per vertex, is 
. In a regular polygon, all of these angles are congruent, so divide 360 by 17 to get the measure of one exterior angle:
The sum of the measures of the exterior angles of any polygon, one per vertex, is
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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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What is the measure of one exterior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?
What is the measure of one exterior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?
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The sum of the measures of the exterior angles of any polygon, one per vertex, is 
. In a regular polygon, all of these angles are congruent, so divide 360 by 23 to get the measure of one exterior angle:
The sum of the measures of the exterior angles of any polygon, one per vertex, is
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What is the measure of one interior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?
What is the measure of one interior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?
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The measure of each interior angle of a regular polygon with 
sides is 
. We can substitute 
to obtain the angle measure:
The measure of each interior angle of a regular polygon with
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What is the magnitude of the interior angle of a regular nonagon?
What is the magnitude of the interior angle of a regular nonagon?
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The equation to calculate the magnitude of an interior angle is 
, where 
is equal to the number of sides.
For our question, 
.
The equation to calculate the magnitude of an interior angle is
For our question,
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What is the interior angle measure of any regular heptagon?
What is the interior angle measure of any regular heptagon?
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To find the angle of any regular polygon you find the number of sides, 
. In this example, 
.
You then subtract 2 from the number of sides yielding 5.
Take 5 and multiply it by 180 degrees to yield the total number of degrees in the regular heptagon. 
Then to find one individual angle we divide 900 by the total number of angles, 7.
The answer is 
.
To find the angle of any regular polygon you find the number of sides,
You then subtract 2 from the number of sides yielding 5.
Take 5 and multiply it by 180 degrees to yield the total number of degrees in the regular heptagon.
Then to find one individual angle we divide 900 by the total number of angles, 7.
The answer is
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