HSPT Quantitative : How to make geometric comparisons
Study concepts, example questions & explanations for HSPT Quantitative
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Example Questions
Example Question #21 : How To Make Geometric Comparisons
Examine (a), (b), and (c) and find the best answer.
a) The area of a square with a side length of
b) The area of a square with a side length of
c) The area of a circle with a radius of
a = b < c
a = c < b
c < b < a
c > b > a
c > b > a
a) The area of a square with a side length of
To find the area of a square, square the side length:
b) The area of a square with a side length of
c) The area of a circle with a radius of
To find the area of a circle, multiply the radius by 
Therefore (c) is larger than (b) which is larger than (a).
Example Question #22 : How To Make Geometric Comparisons
Examine (a), (b), and (c) to find the best answer:
a) the interior angle of an equilateral triangle
b) the interior angle of a square
c) the interior angle of a regular pentagon
Since the interior angles of a triangle add up to 
Each of the interior angles of a square is 
The interior angles of a pentagon add up to 
Example Question #23 : How To Make Geometric Comparisons
Examine (a), (b), and (c) to find the best answer:
a) half the volume of a cube with dimensions 
b) the volume of a cube with dimensions 
c) the volume of a cube with dimensions 
(a), (b), and (c) are all unequal
(a) is equal to (b) but not (c)
(a), (b), and (c) are all equal
(a) is equal to (c) but not (b)
(a) is equal to (b) but not (c)
Find the three volume by multiplying height by length by width:
a)
Half of this volume is 
b)
c)
Remember that we are only looking at half of the volume in a).
Therefore (a) and (b) are equal but (c) is not.
Example Question #23 : How To Make Geometric Comparisons
Examine (a), (b), and (c) to find the best answer:
a) area of a rectangle with side lengths 
b) area of a rectangle with side lengths 
c) area of a square with side length
Area is calculated by multiplying the side lengths:
a) area of a rectangle with side lengths 
b) area of a rectangle with side lengths 
c) area of a square with side length
Therefore (b) is less than (a), which is less than (c).
Example Question #24 : How To Make Geometric Comparisons
Examine (a), (b), and (c) to find the best answer:
a) side length of a cube with a volume of 
b) side length of a square with an area of 
c) side length of a square with an area of 
To find the side length of a cube from its volume, find the cube root:
To find the side length of a square from its area, find the square root:
b)
c)
(a) is smaller than (c), which is smaller than (b)
Example Question #26 : How To Make Geometric Comparisons
What are the relationships between the areas of these shapes?
a. A circle with radius
b. A square with side
c. A rectangle with side lengths of 
First, we find the areas of a, b, and c.
Now we put them in order of size.
Example Question #27 : How To Make Geometric Comparisons
Find the relationship between the perimeters of these shapes.
a. A square with area
b. A circle with diameter
c. A pentagon with side length
First, find the perimeter of the shapes.
Since the area of 
The perimeter of 
The perimeter of 
Since 
Therefore, 
Example Question #27 : How To Make Geometric Comparisons
Find the relationship between these lengths.
a. Side of a square of area
b. Side of a square with perimeter
c. Diameter of a circle with area
First, find the lengths given.
Since the square in 
All sides of a square have equal lengths, so 
The area of a circle is 
The three lengths are equal, so 
Example Question #28 : How To Make Geometric Comparisons
Find the relationship between the areas of the following shapes.
a. Square with perimeter
b. Triangle with base 
c. Circle with circumference
First, find the areas.
Since the perimeter of 
For triangles, 
For 
Putting them in order, we get 
Example Question #29 : How To Make Geometric Comparisons
Find the relationship between the perimeters of the following shapes.
a. Regular hexagon with side length
b. Square with side length
c. Equilateral triangle with side length
Find the perimeters first.
Putting them in order of size:
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