HSPT Quantitative : How to make geometric comparisons
Study concepts, example questions & explanations for HSPT Quantitative
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Example Questions
Example Question #1 : Geometric Comparison
Examine (a), (b), and (c) to find the best answer:
a) area of a circle with a diameter of
b) area of a circle with a radius of
c) area of a circle with a cirucumference of
All of these circles have the same diameter, so they all must have the same area:
a) area of a circle with a diameter of
b) area of a circle with a radius of
c) area of a circle with a cirucumference of
Example Question #2 : Geometric Comparison
Examine (a), (b), and (c) to find the best answer:
a) perimeter of a square with a side length of
b) perimeter of a rectangle with a length of 
c) perimeter of an equailateral triangle with a side length of
To find perimeter, add up the lengths of all the sides:
a)
b)
c)
(a) and (b) are equal, and they are smaller than (c)
Example Question #2 : Geometric Comparison
Examine (a), (b), and (c) to find the best answer:
a) circumference of a circle with a radius of
b) circumference of a circle with a diameter of
c) diameter of circle with a circumference of
The formulas to remember for this problem are 
a) circumference of a circle with a radius of
b) circumference of a circle with a diameter of
c) diameter of circle with a circumference of
Therefore (a) is the greatest, followed by (b), then (c).
Example Question #3 : Geometric Comparison
Examine (a), (b), and (c) to find the best answer:
a) the sum of the interior angles of a triangle
b) the sum of the interior angles of a square
c) the total degrees in a circle
(b) and (c) are equal
(a) and (b) are equal
(a), (b), and (c) are all equal
(c) is greater than (a) and (b)
(b) and (c) are equal
The sum of the interior angles of a triangle is always 
Example Question #4 : Geometric Comparison
Examine (a), (b), and (c) to find the best answer:
a) area of a square with side length
b) area of a square with perimeter
c) area of a square with side length
(a) and (c) are equal
(b) and (c) are equal
(a), (b), and (c) are all equal
(b) is greater than (a) or (c)
(b) and (c) are equal
The area of a square is the side length squared. The perimeter is the side length multiplied by 
(b) and (c) are equal because a side length should be 
Example Question #6 : Geometric Comparison
Examine (a), (b), and (c) to find the best answer:
a) area of a square with a perimeter of
b) area of a square with a side length of
c) area of a square with a side length of
(a) is greater than (b) and (c).
(c) is greater than (a) and (b).
(a) and (b) are equal.
(a) and (c) are equal.
(c) is greater than (a) and (b).
Things to remember here are that area is found by squaring side length and that side length is 
a)
b)
c)
(c) is the greatest, and none of the values are equal.
Example Question #5 : Geometric Comparison
Examine (a), (b), and (c) to find the best answer
a) a square with an area of
b) a square with a side length of
c) a square with a perimeter of
All of these squares are equal! We can tell because they all have the same side length. For (a), find the square root of the area to find the side length:
For (c), divide the perimeter by four to find the side length:
For (b), we are told that the side length equals 
Example Question #8 : Geometric Comparison
Examine (a), (b), and (c) to find the best answer:
a) a circle with a radius of
b) a circle with a diameter of
c) a circle with a circumference of
(a), (b), and (c) have equal area.
(a) has the greatest area.
(c) has the greatest area.
(b) has the greatest area.
(c) has the greatest area.
The circle with the greatest radius is also going to have the greatest area, because 
Use these formulas to find the radius of each circle:
Compare these to (a), with 
Therefore (c) has the largest radius, so it also has the largest area.
Example Question #6 : Geometric Comparison
Examine (a), (b), and (c) to find the best answer:
a) a circle with a circumference of
b) a circle with a radius of
c) a circle with a radius of
(a), (b) and (c) are all equal.
(a), (b) and (c) are all unequal.
(a) is equal to (c) but not (b).
(a) is equal to (b) but not (c).
(a) is equal to (c) but not (b).
Circumference is found by multiplying the diameter by pi, so the diameter of (a) must be 
Example Question #10 : Geometric Comparison
Examine (a), (b), and (c) to find the best answer:
a) a circle with a radius of
b) a circle with a radius of
c) a circle with an area of
(a) is equivalent to (c) but not (b)
(a), (b), and (c) are all not equivalent
(a), (b), and (c) are all equivalent
(b) is equivalent to (c) but not (a)
(b) is equivalent to (c) but not (a)
Find the radius of (c) to compare it to (a) and (b).
Since area is 
Since the radius of (c) is equal to the radius of (b), the circles are equivalent.
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