Quadrilaterals - SAT Math
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If a polygon has 10 sides, what is the measure of each exterior angle?
If a polygon has 10 sides, what is the measure of each exterior angle?
In order to figure this out, we need to remember the formula for finding exterior angles. 
, where 
is the number of sides of a polygon. Now we simply do the following calculation.
So the measure of each exterior angle is 
.
In order to figure this out, we need to remember the formula for finding exterior angles.
So the measure of each exterior angle is
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Quadrilateral ABCD contains four ninety-degree angles. Which of the following must be true?
I. Quadrilateral ABCD is a rectangle.
II. Quadrilateral ABCD is a rhombus.
III. Quadrilateral ABCD is a square.
Quadrilateral ABCD contains four ninety-degree angles. Which of the following must be true?
I. Quadrilateral ABCD is a rectangle.
II. Quadrilateral ABCD is a rhombus.
III. Quadrilateral ABCD is a square.
Quadrilateral ABCD has four ninety-degree angles, which means that it has four right angles because every right angle measures ninety degrees. If a quadrilateral has four right angles, then it must be a rectangle by the definition of a rectangle. This means statement I is definitely true.
However, just because ABCD has four right angles doesn't mean that it is a rhombus. In order for a quadrilateral to be considered a rhombus, it must have four congruent sides. It's possible to have a rectangle whose sides are not all congruent. For example, if a rectangle has a width of 4 meters and a length of 8 meters, then not all of the sides of the rectangle would be congruent. In fact, in a rectangle, only opposite sides need be congruent. This means that ABCD is not necessarily a rhombus, and statement II does not have to be true.
A square is defined as a rhombus with four right angles. In a square, all of the sides must be congruent. In other words, a square is both a rectangle and a rhombus. However, we already established that ABCD doesn't have to be a rhombus. This means that ABCD need not be a square, because, as we said previously, not all of its sides must be congruent. Therefore, statement III isn't necessarily true either.
The only statement that has to be true is statement I.
The answer is I only.
Quadrilateral ABCD has four ninety-degree angles, which means that it has four right angles because every right angle measures ninety degrees. If a quadrilateral has four right angles, then it must be a rectangle by the definition of a rectangle. This means statement I is definitely true.
However, just because ABCD has four right angles doesn't mean that it is a rhombus. In order for a quadrilateral to be considered a rhombus, it must have four congruent sides. It's possible to have a rectangle whose sides are not all congruent. For example, if a rectangle has a width of 4 meters and a length of 8 meters, then not all of the sides of the rectangle would be congruent. In fact, in a rectangle, only opposite sides need be congruent. This means that ABCD is not necessarily a rhombus, and statement II does not have to be true.
A square is defined as a rhombus with four right angles. In a square, all of the sides must be congruent. In other words, a square is both a rectangle and a rhombus. However, we already established that ABCD doesn't have to be a rhombus. This means that ABCD need not be a square, because, as we said previously, not all of its sides must be congruent. Therefore, statement III isn't necessarily true either.
The only statement that has to be true is statement I.
The answer is I only.
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Find the area of a parallelogram with length 6 and height 5.
Find the area of a parallelogram with length 6 and height 5.
To solve, simply use the formula for the area of a parallelogram. Thus,
Don't let a parallelogram fool you. It is just like a rectangle but on a slant. As long as you are given the height, and not the slant side length, you can use the same formula.
To solve, simply use the formula for the area of a parallelogram. Thus,
Don't let a parallelogram fool you. It is just like a rectangle but on a slant. As long as you are given the height, and not the slant side length, you can use the same formula.
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Find the perimeter of a parallelogram with one side equal to 6 and the other to 5.
Find the perimeter of a parallelogram with one side equal to 6 and the other to 5.
To solve, simply use the formula for the perimeter of a parallelogram. Thus,
If you don't remember this formula, you can also just sum up all the sides of the parallelogram.
Where s1 and s3 are the long sides and s2 and s4 are the short sides. Since the two pairs of sides are equal, you can sub in the following.
To solve, simply use the formula for the perimeter of a parallelogram. Thus,
If you don't remember this formula, you can also just sum up all the sides of the parallelogram.
Where s1 and s3 are the long sides and s2 and s4 are the short sides. Since the two pairs of sides are equal, you can sub in the following.
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George wants to paint the walls in his room blue. The ceilings are 10 ft tall and a carpet 12 ft by 15 ft covers the floor. One gallon of paint covers 400 
and costs $40. One quart of paint covers 100 
and costs $15. How much money will he spend on the blue paint?
George wants to paint the walls in his room blue. The ceilings are 10 ft tall and a carpet 12 ft by 15 ft covers the floor. One gallon of paint covers 400
The area of the walls is given by 
One gallon of paint covers 400 
and the remaining 140 
would be covered by two quarts.
So one gallon and two quarts of paint would cost 
The area of the walls is given by
One gallon of paint covers 400
So one gallon and two quarts of paint would cost
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A contractor is going to re-tile a rectangular section of the kitchen floor. If the floor is 6ft x 3ft, and he is going to use square tiles with a side of 9in. How many tiles will be needed?
A contractor is going to re-tile a rectangular section of the kitchen floor. If the floor is 6ft x 3ft, and he is going to use square tiles with a side of 9in. How many tiles will be needed?
We have to be careful of our units. The floor is given in feet and the tile in inches. Since the floor is 6ft x 3ft. we can say it is 72in x 36in, because 12 inches equals 1 foot. If the tiles are 9in x 9in we can fit 8 tiles along the length and 4 tiles along the width. To find the total number of tiles we multiply 8 x 4 = 32. Alternately we could find the area of the floor (72 x 36, and divide by the area of the tile 9 x 9)
We have to be careful of our units. The floor is given in feet and the tile in inches. Since the floor is 6ft x 3ft. we can say it is 72in x 36in, because 12 inches equals 1 foot. If the tiles are 9in x 9in we can fit 8 tiles along the length and 4 tiles along the width. To find the total number of tiles we multiply 8 x 4 = 32. Alternately we could find the area of the floor (72 x 36, and divide by the area of the tile 9 x 9)
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If the area Rectangle A is 
larger than Rectangle B and the sides of Rectangle A are 
and 
, what is the area of Rectangle B?
If the area Rectangle A is
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The front façade of a building is 100 feet tall and 40 feet wide. There are eight floors in the building, and each floor has four glass windows that are 8 feet wide and 6 feet tall along the front façade. What is the total area of the glass in the façade?
The front façade of a building is 100 feet tall and 40 feet wide. There are eight floors in the building, and each floor has four glass windows that are 8 feet wide and 6 feet tall along the front façade. What is the total area of the glass in the façade?
Glass Area per Window = 8 ft x 6 ft = 48 ft2
Total Number of Windows = Windows per Floor * Number of Floors = 4 * 8 = 32 windows
Total Area of Glass = Area per Window * Total Number of Windows = 48 * 32 = 1536 ft2
Glass Area per Window = 8 ft x 6 ft = 48 ft2
Total Number of Windows = Windows per Floor * Number of Floors = 4 * 8 = 32 windows
Total Area of Glass = Area per Window * Total Number of Windows = 48 * 32 = 1536 ft2
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The rectangular bathroom floor in Michael’s house is ten feet by twelve feet. He wants to purchase square tiles that are four inches long and four inches wide to cover the bathroom floor. If each square tile costs $2.50, how much money will Michael need to spend in order to purchase enough tiles to cover his entire bathroom floor?
The rectangular bathroom floor in Michael’s house is ten feet by twelve feet. He wants to purchase square tiles that are four inches long and four inches wide to cover the bathroom floor. If each square tile costs $2.50, how much money will Michael need to spend in order to purchase enough tiles to cover his entire bathroom floor?
The dimensions for the bathroom are given in feet, but the dimensions of the tiles are given in inches; therefore, we need to convert the dimensions of the bathroom from feet to inches, because we can’t compare measurements easily unless we are using the same type of units.
Because there are twelve inches in a foot, we need to multiply the number of feet by twelve to convert from feet to inches.
10 feet = 10 x 12 inches = 120 inches
12 feet = 12 x 12 inches = 144 inches
This means that the bathroom floor is 120 inches by 144 inches. The area of Michael’s bathroom is therefore 120 x 144 in2 = 17280 in2.
Now, we need to find the area of the tiles in square inches and calculate how many tiles it would take to cover 17280 in2.
Each tile is 4 in by 4 in, so the area of each tile is 4 x 4 in2, or 16 in2.
If there are 17280 in2 to be covered, and each tile is 16 in2, then the number of tiles we need is 17280 ÷ 16, which is 1080 tiles.
The question ultimately asks us for the cost of all these tiles; therefore, we need to multiply 1080 by 2.50, which is the price of each tile.
The total cost = 1080 x 2.50 dollars = 2700 dollars.
The answer is $2700.
The dimensions for the bathroom are given in feet, but the dimensions of the tiles are given in inches; therefore, we need to convert the dimensions of the bathroom from feet to inches, because we can’t compare measurements easily unless we are using the same type of units.
Because there are twelve inches in a foot, we need to multiply the number of feet by twelve to convert from feet to inches.
10 feet = 10 x 12 inches = 120 inches
12 feet = 12 x 12 inches = 144 inches
This means that the bathroom floor is 120 inches by 144 inches. The area of Michael’s bathroom is therefore 120 x 144 in2 = 17280 in2.
Now, we need to find the area of the tiles in square inches and calculate how many tiles it would take to cover 17280 in2.
Each tile is 4 in by 4 in, so the area of each tile is 4 x 4 in2, or 16 in2.
If there are 17280 in2 to be covered, and each tile is 16 in2, then the number of tiles we need is 17280 ÷ 16, which is 1080 tiles.
The question ultimately asks us for the cost of all these tiles; therefore, we need to multiply 1080 by 2.50, which is the price of each tile.
The total cost = 1080 x 2.50 dollars = 2700 dollars.
The answer is $2700.
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Ron has a fixed length of wire that he uses to make a lot. On Monday, he uses the wire to make a rectangular lot. On Tuesday, he uses the same length of wire to form a square-shaped lot. Ron notices that the square lot has slightly more area, and he determines that the difference between the areas of the two lots is sixteen square units. What is the positive difference, in units, between the length and the width of the lot on Monday?
Ron has a fixed length of wire that he uses to make a lot. On Monday, he uses the wire to make a rectangular lot. On Tuesday, he uses the same length of wire to form a square-shaped lot. Ron notices that the square lot has slightly more area, and he determines that the difference between the areas of the two lots is sixteen square units. What is the positive difference, in units, between the length and the width of the lot on Monday?
Let’s say that the rectangular lot on Monday has a length of l and a width of w. The area of a rectangular is the product of the length and the width, so we can write the area of the lot on Monday as lw.
Next, we need to find an expression for the area of the lot on Tuesday. We are told that the lot is in the shape of a square and that it uses the same length of wire. If the length of the wire used is the same on both days, then the perimeter will have to remain the same. In other words, the perimeter of the square will equal the perimeter of the rectangle. The perimeter of a rectangle is given by 2_l_ + 2_w_.
We also know that if s is the length of a side of a square, then the perimeter is 4_s_, because each side of the square is congruent. Let’s write an equation that sets the perimeter of the rectangle and the square equal.
2_l_ + 2_w_ = 4_s_
If we divide both sides by 4 and then simplify the expression, then we can write the length of the square as follows:
Let’s say that the rectangular lot on Monday has a length of l and a width of w. The area of a rectangular is the product of the length and the width, so we can write the area of the lot on Monday as lw.
Next, we need to find an expression for the area of the lot on Tuesday. We are told that the lot is in the shape of a square and that it uses the same length of wire. If the length of the wire used is the same on both days, then the perimeter will have to remain the same. In other words, the perimeter of the square will equal the perimeter of the rectangle. The perimeter of a rectangle is given by 2_l_ + 2_w_.
We also know that if s is the length of a side of a square, then the perimeter is 4_s_, because each side of the square is congruent. Let’s write an equation that sets the perimeter of the rectangle and the square equal.
2_l_ + 2_w_ = 4_s_
If we divide both sides by 4 and then simplify the expression, then we can write the length of the square as follows:
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A rectangle has a width of 2_x_. If the length is five more than 150% of the width, what is the area of the rectangle?
A rectangle has a width of 2_x_. If the length is five more than 150% of the width, what is the area of the rectangle?
Given that w = 2_x_ and l = 1.5_w_ + 5, a substitution will show that l = 1.5(2_x_) + 5 = 3_x_ + 5.
A = lw = (3_x_ + 5)(2_x_) = 6_x_2 + 10_x_
Given that w = 2_x_ and l = 1.5_w_ + 5, a substitution will show that l = 1.5(2_x_) + 5 = 3_x_ + 5.
A = lw = (3_x_ + 5)(2_x_) = 6_x_2 + 10_x_
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Rectangle ABCD is shown in the figure above. Points A and B lie on the graph of y = 64 – _x_2 , and points C and D lie on the graph of y = _x_2 – 36. Segments AD and BC are both parallel to the y-axis. The x-coordinates of points A and B are equal to –k and k, respectively. If the value of k changes from 2 to 4, by how much will the area of rectangle ABCD increase?
Rectangle ABCD is shown in the figure above. Points A and B lie on the graph of y = 64 – _x_2 , and points C and D lie on the graph of y = _x_2 – 36. Segments AD and BC are both parallel to the y-axis. The x-coordinates of points A and B are equal to –k and k, respectively. If the value of k changes from 2 to 4, by how much will the area of rectangle ABCD increase?
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Two circles of a radius of 
each sit inside a square with a side length of 
. If the circles do not overlap, what is the area outside of the circles, but within the square?
Two circles of a radius of
The area of a square = 
The area of a circle is 
Area = Area of Square 
2(Area of Circle) =
The area of a square =
The area of a circle is
Area = Area of Square
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Daisy gets new carpet for her rectangluar room. Her floor is 
. The carpet sells for $5 per square yard. How much did she spend on her carpet?
Daisy gets new carpet for her rectangluar room. Her floor is
Since 
the room measurements are 7 yards by 8 yards. The area of the floor is thus 56 square yards. It would cost 
.
Since
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The length of a rectangular rug is five more than twice its width. The perimeter of the rug is 40 ft. What is the area of the rug?
The length of a rectangular rug is five more than twice its width. The perimeter of the rug is 40 ft. What is the area of the rug?
For a rectangle, 
and 
where 
is the width and 
is the length.
Let 
and 
.
So the equation to solve becomes 
or 
.
Thus 
and 
, so the area is 
.
For a rectangle,
Let
So the equation to solve becomes
Thus
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The width of a rectangle is 
. The length of the rectangle is 
. What must be the area?
The width of a rectangle is
The area of a rectangle is:
Substitute the variables into the formula.
The area of a rectangle is:
Substitute the variables into the formula.
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Find the area of a rectangle with side length 7 and 9.
Find the area of a rectangle with side length 7 and 9.
To solve, simply use the formula for the area of a rectangle.
Substitute in the side length of 7 and width of 9.
Thus,
To solve, simply use the formula for the area of a rectangle.
Substitute in the side length of 7 and width of 9.
Thus,
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Find the area of a rectanlge given width is 2 and length is 3.
Find the area of a rectanlge given width is 2 and length is 3.
To solve, simply use the formula for the area of a rectangle. Thus,
To solve, simply use the formula for the area of a rectangle. Thus,
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A parallelogram with right angles has side lengths of 
and 
. What is its area?
A parallelogram with right angles has side lengths of
Remember that a parallelogram with right angles is a rectangle. With that in mind, all you need to do is multiply those side lengths together, knowing that they are the length and width of a rectangle: 
Remember that a parallelogram with right angles is a rectangle. With that in mind, all you need to do is multiply those side lengths together, knowing that they are the length and width of a rectangle:
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Find the area of a rectangle given width 6 and length 9.
Find the area of a rectangle given width 6 and length 9.
To solve, simply multiply the width by the length. Using the formula, you get the answer as follows:
Additionally, you can alternatively solve this problem by drawing out a rectangle, creating 6 horizontal lines and 9 vertical ones, and then adding up the squares to reach your answer.
To solve, simply multiply the width by the length. Using the formula, you get the answer as follows:
Additionally, you can alternatively solve this problem by drawing out a rectangle, creating 6 horizontal lines and 9 vertical ones, and then adding up the squares to reach your answer.
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