Geometry
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ISEE Upper Level Quantitative Reasoning › Geometry
Find the perimeter of a pentagon with a side of length 12in.
Explanation
To find the perimeter of a pentagon, we will use the following formula:
where a is the length of one side of the pentagon.
Now, we know the length of one side of the pentagon is 12in.
Knowing this, we can substitute into the formula. We get
Give the area of the white region of the above circle if 
Explanation
If we let 
The radius is the circumference divided by 
Use the formula to find the area of the entire circle:
The area of the white region is 
A wooden ball has a surface area of 
What is its radius?
Cannot be determined from the information provided
Explanation
A wooden ball has a surface area of 
What is its radius?
Begin with the formula for surface area of a sphere:
Now, plug in our surface area and solve with algebra:
Get rid of the pi
Divide by 4
Square root both sides to get our answer:
A cube has a side length of 
Explanation
A cube has a side length of 
To find the volume of a cube, use the following formula:
Plug in our known side length and solve
Making our answer:
A square is made into a rectangle by increasing the width by 20% and decreasing the length by 20%. By what percentage has the area of the square changed?
decreased by 4%
increased by 20%
the area remains the same
decreased by 10%
Explanation
The area decreases by 20% of 20%, which is 4%.
The easiest way to see this is to plug in numbers for the sides of the square. If we are using percentages, it is easiest to use factors of 10 or 100. In this case we will say that the square has a side length of 10.
10% of 10 is 1, so 20% is 2. Now we can just increase one of the sides by 2, and decrease another side by 2. So our rectangle has dimensions of 12 x 8 instead of 10 x 10.
The original square had an area of 100, and the new rectangle has an area of 96. So the rectangle is 4 square units smaller, which is 4% smaller than the original square.
If a cube has one side measuring 
Explanation
To find the surface area of a cube, use the formula 
If a cube has one side measuring
Explanation
To find the surface area of a cube, use the formula
A square is made into a rectangle by increasing the width by 20% and decreasing the length by 20%. By what percentage has the area of the square changed?
decreased by 4%
increased by 20%
the area remains the same
decreased by 10%
Explanation
The area decreases by 20% of 20%, which is 4%.
The easiest way to see this is to plug in numbers for the sides of the square. If we are using percentages, it is easiest to use factors of 10 or 100. In this case we will say that the square has a side length of 10.
10% of 10 is 1, so 20% is 2. Now we can just increase one of the sides by 2, and decrease another side by 2. So our rectangle has dimensions of 12 x 8 instead of 10 x 10.
The original square had an area of 100, and the new rectangle has an area of 96. So the rectangle is 4 square units smaller, which is 4% smaller than the original square.
Three of the vertices of a parallelogram on the coordinate plane are 
Insufficient information is given to answer the problem.
Explanation
As can be seen in the diagram, there are three possible locations of the fourth point of the parallelogram:
Regardless of the location of the fourth point, however, the triangle with the given three vertices comprises exactly half the parallelogram. Therefore, the parallelogram has double that of the triangle.
The area of the triangle can be computed by noting that the triangle is actually a part of a 12-by-12 square with three additional right triangles cut out:
The area of the 12 by 12 square is
The area of the green triangle is 
The area of the blue triangle is 
The area of the pink triangle is 
The area of the main triangle is therefore
The parallelogram has area twice this, or 
If a cube has one side measuring
Explanation
To find the surface area of a cube, use the formula