Plane Geometry - ISEE Middle Level Quantitative Reasoning
Card 0 of 2145
Calvin is remodeling his room. He used 
feet of molding to put molding around all four walls. Now he wants to paint three of the walls. Each wall is the same width and is 
feet tall. If one can of paint covers 
square feet, how many cans of paint will he need to paint three walls.
Calvin is remodeling his room. He used
When Calvin put up 
feet of molding, he figured out the perimeter of the room was 
feet. Since he knows that all four walls are the same width, he can use the equation 
to determine the length of each side by plugging 
in for 
and solving for 
.
In order to solve for 
, Calvin must divide both sides by four.
The left-hand side simplifies to:
The right-hand side simplifies to:
Now, Calvin knows the width of each room is 
feet. Next he must find the area of each wall. To do this, he must multiply the width by the height because the area of a rectangle is found using the equation 
. Since Calvin now knows that the width of each wall is 
feet and that the height of each wall is also 
feet, he can multiply the two together to find the area.
Since Calvin wants to find how much paint he needs to cover three walls, he must first find out how many square feet he is covering. If one wall is 
square feet, he must multiply that by 
.
Calvin is painting 
square feet. If one can of paint covers 24 square feet, he must divide the total space (
square feet) by 
.
Calvin will need 
cans of paint.
When Calvin put up
In order to solve for
The left-hand side simplifies to:
The right-hand side simplifies to:
Now, Calvin knows the width of each room is
Since Calvin wants to find how much paint he needs to cover three walls, he must first find out how many square feet he is covering. If one wall is
Calvin is painting
Calvin will need
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Which is the greater quantity?
(a) The surface area of a cube with volume 
(b) The surface area of a cube with sidelength 
Which is the greater quantity?
(a) The surface area of a cube with volume
(b) The surface area of a cube with sidelength
We can actually solve this by comparing volumes; the cube with the greater volume has the greater sidelength and, subsequently, the greater surface area.
The volume of the cube in (b) is the cube of 90 millimeters, or 9 centimeters. This is 
, which is greater than 
. The cube in (b) has the greater volume, sidelength, and, most importantly, surface area.
We can actually solve this by comparing volumes; the cube with the greater volume has the greater sidelength and, subsequently, the greater surface area.
The volume of the cube in (b) is the cube of 90 millimeters, or 9 centimeters. This is
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Each side of a square is 
units long. Which is the greater quantity?
(A) The area of the square
(B) 
Each side of a square is
(A) The area of the square
(B)
The area of a square is the square of its side length:
Using the side length from the question:
However, it is impossible to tell with certainty which of 
and 
is greater.
For example, if 
,
and
so 
if .
But if 
,
and
so 
if .
The area of a square is the square of its side length:
Using the side length from the question:
However, it is impossible to tell with certainty which of
For example, if
and
so
But if
and
so
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The sum of the lengths of three sides of a square is one yard. Give its area in square inches.
The sum of the lengths of three sides of a square is one yard. Give its area in square inches.
A square has four sides of the same length.
One yard is equal to 36 inches, so each side of the square has length
inches.
Its area is the square of the sidelength, or
square inches.
A square has four sides of the same length.
One yard is equal to 36 inches, so each side of the square has length
Its area is the square of the sidelength, or
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The sum of the lengths of three sides of a square is 3,900 centimeters. Give its area in square meters.
The sum of the lengths of three sides of a square is 3,900 centimeters. Give its area in square meters.
100 centimeters are equal to one meter, so 3,900 centimeters are equal to
meters.
A square has four sides of the same length. Since the sum of the lengths of three of the congruent sides is 3,900 centimeters, or 39 meters, each side measures
meters.
The area of the square is the square of the sidelength, or
square meters.
100 centimeters are equal to one meter, so 3,900 centimeters are equal to
A square has four sides of the same length. Since the sum of the lengths of three of the congruent sides is 3,900 centimeters, or 39 meters, each side measures
The area of the square is the square of the sidelength, or
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A square has a side with a length of 5. What is the area of the square?
A square has a side with a length of 5. What is the area of the square?
The area formula for a square is length times width. Keep in mind that all of a square's sides are equal.
So, if one side of a square equals 5, all of the other sides must also equal 5. You will find the area of the square by multiplying two of its sides:
The area formula for a square is length times width. Keep in mind that all of a square's sides are equal.
So, if one side of a square equals 5, all of the other sides must also equal 5. You will find the area of the square by multiplying two of its sides:
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One square mile is equivalent to 640 acres. Which of the following is the greater quantity?
(a) The area of a square plot of land whose perimeter measures one mile
(b) 160 acres
One square mile is equivalent to 640 acres. Which of the following is the greater quantity?
(a) The area of a square plot of land whose perimeter measures one mile
(b) 160 acres
A square plot of land with perimeter one mile has as its sidelength one fourth of this, or 
mile; its area is the square of this, or
square miles.
One square mile is equivalent to 640 acres, so 
square miles is equivalent to
acres.
This makes (b) greater.
A square plot of land with perimeter one mile has as its sidelength one fourth of this, or
One square mile is equivalent to 640 acres, so
This makes (b) greater.
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One square kilometer is equal to 100 hectares.
Which is the greater quantity?
(a) The area of a rectangular plot of land 500 meters in length and 200 meters in width
(b) One hectare
One square kilometer is equal to 100 hectares.
Which is the greater quantity?
(a) The area of a rectangular plot of land 500 meters in length and 200 meters in width
(b) One hectare
One kilometer is equal to 1,000 meters, so divide each dimension of the plot in meters by 1,000 to convert to kilometers:
kilometers
kilometers
Multiply the dimensions to get the area in square kilometers:
square kilometers
Since one square kilometer is equal to 100 hectares, multiply this by 100 to convert to hectares:
hectares
This makes (a) the greater.
One kilometer is equal to 1,000 meters, so divide each dimension of the plot in meters by 1,000 to convert to kilometers:
Multiply the dimensions to get the area in square kilometers:
Since one square kilometer is equal to 100 hectares, multiply this by 100 to convert to hectares:
This makes (a) the greater.
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Using the information given in each question, compare the quantity in Column A to the quantity in Column B.
A certain rectangle is seven times as long as it is wide.
Column A Column B
the rectangle's the rectangle's
perimeter area
(in units) (in square units)
Using the information given in each question, compare the quantity in Column A to the quantity in Column B.
A certain rectangle is seven times as long as it is wide.
Column A Column B
the rectangle's the rectangle's
perimeter area
(in units) (in square units)
This type of problem reminds us to be wary of simply plugging in numbers (which works with certain problems). If you were to choose 1 and 7 here, the perimeter would be larger; if you chose 10 and 70, the area would be much larger.
To solve this problem with variables:
From here we can see that smaller values of 
will lead to a larger perimeter, while larger values of 
will lead to a larger area.
The answer cannot be determined.
This type of problem reminds us to be wary of simply plugging in numbers (which works with certain problems). If you were to choose 1 and 7 here, the perimeter would be larger; if you chose 10 and 70, the area would be much larger.
To solve this problem with variables:
From here we can see that smaller values of
The answer cannot be determined.
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If the length of a rectangle is twice the width, and the width is three inches, what is the area of the rectangle?
If the length of a rectangle is twice the width, and the width is three inches, what is the area of the rectangle?
In order to find the area of a rectangle we use the formula 
.
In this problem, we know the width is 
. We also know that the length is twice as long as the width, which can be written as 
. This means that in order to find the length, we must multiply the width by 
.
Now that we know that our length is 
, we simply multiply it by our width of 
.
The area of the rectangle is 
.
In order to find the area of a rectangle we use the formula
In this problem, we know the width is
Now that we know that our length is
The area of the rectangle is
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Which is the greater quantity?
(a) The surface area of a rectangular prism with length 60 centimeters, width 30 centimeters, and height 15 centimeters
(b) The surface area of a cube with sidelength 300 millimeters
Which is the greater quantity?
(a) The surface area of a rectangular prism with length 60 centimeters, width 30 centimeters, and height 15 centimeters
(b) The surface area of a cube with sidelength 300 millimeters
(a) The surface of a rectangular prism comprises six rectangles, so we can take the sum of their areas.
Two rectangles have area: 
.
Two rectangles have area: 
.
Two rectangles have area: 
.
Add the areas: 
(b) The surface of a cube comprises six squares, so we can square the sidelength - which we rewrite as 30 centimeters - and multiply the result by 6:
.
The first figure has the greater surface area.
(a) The surface of a rectangular prism comprises six rectangles, so we can take the sum of their areas.
Two rectangles have area:
Two rectangles have area:
Two rectangles have area:
Add the areas:
(b) The surface of a cube comprises six squares, so we can square the sidelength - which we rewrite as 30 centimeters - and multiply the result by 6:
The first figure has the greater surface area.
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Column A Column B
The area of a The area of a square
rectangle with sides with sides 7cm.
11 cm and 5 cm.
Column A Column B
The area of a The area of a square
rectangle with sides with sides 7cm.
11 cm and 5 cm.
First, you must calculate Column A. The formula for the area of a rectangle is 
. Plug in the values given to get 
, which gives you 
. Then, calculate the area of the square. Since all of the sides of a square are equal, the formula is 
, or 
. Therefore, the area of the square is 
, which gives you 
. Therefore, the quantity in Column A is greater.
First, you must calculate Column A. The formula for the area of a rectangle is
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What is the length of a rectangular room with an area of 
and a width of 
What is the length of a rectangular room with an area of
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
Compare your answer with the correct one above
What is the length of a rectangular room with an area of 
and a width of 
What is the length of a rectangular room with an area of
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
Compare your answer with the correct one above
What is the length of a rectangular room with an area of 
and a width of 
What is the length of a rectangular room with an area of
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
Compare your answer with the correct one above
What is the length of a rectangular room with an area of 
and a width of 
What is the length of a rectangular room with an area of
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
Compare your answer with the correct one above
What is the length of a rectangular room with an area of 
and a width of 
What is the length of a rectangular room with an area of
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
Compare your answer with the correct one above
What is the length of a rectangular room with an area of 
and a width of 
What is the length of a rectangular room with an area of
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
Compare your answer with the correct one above
What is the length of a rectangular room with an area of 
and a width of 
What is the length of a rectangular room with an area of
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
Compare your answer with the correct one above
What is the length of a rectangular room with an area of 
and a width of 
What is the length of a rectangular room with an area of
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
We have the area and the width, so we can plug those values into our equation and solve for our unknown.
Compare your answer with the correct one above