How to make geometric comparisons - HSPT Quantitative
Card 1 of 164
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 
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
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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 
.
(Here, we rounded 
to 
, because an exact number isn't necessary to answer the question.)
Therefore (c) is larger than (b) which is larger than (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).
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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 
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
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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 
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
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Examine (a), (b), and (c) to find the best answer:
a) area of a rectangle with side lengths 
and 
b) area of a rectangle with side lengths 
and 
c) area of a square with side length 
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
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Area is calculated by multiplying the side lengths:
a) area of a rectangle with side lengths 
and 
b) area of a rectangle with side lengths 
and 
c) area of a square with side length 
Therefore (b) is less than (a), which is less than (c).
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).
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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 
and a width of 
c) perimeter of an equailateral triangle with a side length of 
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
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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)
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)
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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 
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
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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).
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).
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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 
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
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The formulas to remember for this problem are 
(Circumference equals diameter times pi) and 
(diameter equals two pi).
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).
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).
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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 
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
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Things to remember here are that area is found by squaring side length and that side length is 
of the perimeter.
a) 
b) 
c) 
(c) is the greatest, and none of the values are equal.
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.
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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
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
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The sum of the interior angles of a triangle is always 
degrees. For squares, it's always 
degrees. There are also a total of 
degrees in a circle. Therefore, (b) and (c) are equal, and they are greater than (a).
The sum of the interior angles of a triangle is always
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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 
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
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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 
of the perimeter. (a) is the greatest, because the greatest side length leads to the greatest area.
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
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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 
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
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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:
and 
, so in (b), 
.
and 
, so in (c), 
.
Compare these to (a), with 
.
Therefore (c) has the largest radius, so it also has the largest 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.
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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 
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
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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 
.
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
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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 
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
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Circumference is found by multiplying the diameter by pi, so the diameter of (a) must be 
. Radius is half of diameter, so the radius of (a) must be 
. This means that (a) is equal to (c), but not (b).
Circumference is found by multiplying the diameter by pi, so the diameter of (a) must be
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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 
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
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Find the radius of (c) to compare it to (a) and (b).
Since area is 
, we know that 
must be 
and that 
the square root of 
must be 
.
Since the radius of (c) is equal to the radius of (b), the circles are equivalent.
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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Examine (a), (b), and (c) to find the best answer:
a) side length of a cube with a volume of 
inches cubed
b) side length of a square with an area of 
inches squared
c) side length of a square with an area of 
inches squared
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
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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)
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)
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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 
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
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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.
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.
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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 
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
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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 
:
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
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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
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
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The shape of a triangle does not affect its size. Without knowing any dimensions, we cannot determine how large any of the areas are.
The shape of a triangle does not affect its size. Without knowing any dimensions, we cannot determine how large any of the areas are.
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Circle 
has a radius of 
inches, a circumference of 
inches, and an area of 
inches squared.
Now imagine three other circles:
a) Circle 
has a radius of 
b) Circle 
has a circumference of 
c) Circle 
has an area of 
Circle
Now imagine three other circles:
a) Circle
b) Circle
c) Circle
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Radius and circumference are related to each other by a direct proportion (
). This means that if you cut one in half, you cut the other in half. For this reason, circles 
and 
are both related to 
by the same proportion and are thus equivalent.
Area is related to radius exponentially (
). Cutting the area in half does not cut the radius in half. Circle 
therefore cannot be equivalent to 
.
To test this out, try substituting some numbers in for 
, 
, and 
.
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
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Examine (a), (b), and (c) to find the best answer:
a) volume of a cube with side length of 
inches
b) area of a square with a perimeter of 
inches
c) perimeter of a square with a side length of 
inches
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
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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.
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.
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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 
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
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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 
. Since the circumfrence in (a) is the diameter in (c) multiplied by 
, this is also 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
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