HSPT Quantitative : How to make geometric comparisons
Study concepts, example questions & explanations for HSPT Quantitative
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Example Questions
Example Question #11 : Hspt Quantitative Skills
Examine (a), (b), and (c) to find the best answer:
a) the diameter of a circle
b) twice the radius of a circle
c) the circumference of a circle divided by
(a) is equivalent to (c) but not (b)
(a), (b), and (c) are all equivalent
(a), (b), and (c) are all not equivalent
(a) is equivalent to (b) but not (c)
(a), (b), and (c) are all equivalent
Diameter is found using this formula, which shows that it is twice the radius:
Circumference is found using this formula:
Manipulate this to find that diameter is also circumference divided by 
Therefore, all of these expressions are the same.
Example Question #11 : Geometric Comparison
Examine (a), (b), and (c) to find the best answer:
a) perimeter of a square with an area of
b) perimeter of a square with an area of
c) perimeter of a square with a side length of
(c) is two times (b)
(c) is two times (a)
(a) is equal to (c)
(b) is equal to (c)
(c) is two times (b)
If you have the area of a square, you can find the side length by finding the square root. Then multiply by four to find the perimeter:
a)
b)
If you are given the side length, simply multiply by 
Example Question #12 : Hspt Quantitative Skills
Examine (a), (b), and (c) to find the best answer:
(a) area of an equilateral triangle
(b) area of an isoceles triangle
(c) area of a right triangle
It is impossible to say which is greatest.
(a) is the greatest
(c) is the greatest
(b) is the greatest
It is impossible to say which is greatest.
The shape of a triangle does not affect its size. Without knowing any dimensions, we cannot determine how large any of the areas are.
Example Question #12 : Geometric Comparison
Circle 
Now imagine three other circles:
a) Circle 
b) Circle 
c) Circle 
Circle 
Circles 
Circle 
Circles 
Circle
Radius and circumference are related to each other by a direct proportion (
Area is related to radius exponentially (
To test this out, try substituting some numbers in for 
Example Question #13 : Hspt Quantitative Skills
Examine (a), (b), and (c) to find the best answer:
a) volume of a cube with side length of 
b) area of a square with a perimeter of 
c) perimeter of a square with a side length of 
Impossible to compare
Impossible to compare
Voume would be measured in inches cubed, area in inches squared, and perimeter in inches. These are different types of measurements and cannot be compared.
Example Question #14 : Hspt Quantitative Skills
Examine (a), (b), and (c) to find the best answer:
a) area of a circle with a circumference of
b) area of a circle with a radius of
c) area of a circle with a diameter of
(a) is equal to (c) but not (b)
(a), (b), and (c) are all unequal
(b) is equal to (c) but not (a)
(a), (b), and (c) are all equal
(a), (b), and (c) are all equal
The diameter of a circle is twice its radius. Since the diameter in (c) is twice the radius in (b), these two are equal.
The circumference is the diameter multiplied by 
Example Question #17 : Geometric Comparison
Examine (a), (b), and (c) to find the best answer:
a) perimeter of a square with a side length of
b) circumference of a circle with a diameter of
c) perimeter of an equilateral triangle with a side length of
(a) is equal to (c).
(b) is equal to (c).
(b) is greater than (a) and (c).
(a) is greater than (b) and (c).
(a) is equal to (c).
Find perimeter by adding up the lengths of the sides of the shape. Find circumference by multiplying the diameter by 
a) perimeter of a square with a side length of
b) circumference of a circle with a diameter of
c) perimeter of an equilateral triangle with a side length of
Therefore (a) and (c) are equal, and they are greater than (b).
Example Question #11 : How To Make Geometric Comparisons
Examine (a), (b), and (c) to find the best answer:
a) perimeter of an equilateral triangle with a side length of
b) perimeter of a square with an side length of
c) circumference of a circle with a diameter of
Find the perimeter from side length of an equilateral triangle by multiplying by three:
a)
Find perimeter from side length of a square by multiplying by four:
b)
Find circumference from diameter of a circle by multiplying by pi:
c)
It is now evident that (c) is smaller than (a), which is smaller than (b)
Example Question #15 : Hspt Quantitative Skills
Examine (a), (b), and (c) to find the best answer:
a) the sum of the areas of a square and a triangle
b) the sum of the areas of two squares
c) the sum of the areas of two triangles
Not enough information to know
Not enough information to know
We are not given any information about the sizes of the shapes. There could be two tiny squares, one huge triangle, we're not sure. Without any dimensions, we cannot say which area is bigger.
Example Question #13 : How To Make Geometric Comparisons
Compare the values of (a), (b), and (c):
(a) The area of a square with a perimeter of
(b) The area of a square with a side length of
(c) The area of a square with a perimeter of
(b) is equal to (c), but not (a)
(a) is equal to (b), but not (c)
(a), (b), and (c) are all equal
(a), (b), and (c) are all unequal
(a) is equal to (b), but not (c)
Find the areas of all the squares. To find an area from a perimeter, divide the perimeter by 
(a)
(b)
(c)
(a) is equal to (b), but not equal to (c).
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