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HSPT Quantitative : Geometric Comparison

Study concepts, example questions & explanations for HSPT Quantitative

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6 Diagnostic Tests 55 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #21 : Geometric Comparison

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

Possible Answers:

c < b < a

a = c < b

a = b < c

c > b > a

Correct answer:

c > b > a

Explanation:

a) The area of a square with a side length of

To find the area of a square, square the side length: $2^{2}=4$

b) The area of a square with a side length of

$3^{2}=9$

c) The area of a circle with a radius of

To find the area of a circle, multiply the radius by $2\pi$ .

$2\cdot2\pi=4\pi\approx12.56$ (Here, we rounded $\pi$ to 3.14 , because an exact number isn't necessary to answer the question.)

Therefore (c) is larger than (b) which is larger than (a).

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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

Possible Answers:

a>b>c

a<b<c

c<a<b

a=b=c

Correct answer:

a<b<c

Explanation:

Since the interior angles of a triangle add up to 180 , each angle of an equilateral triangle is degrees.

Each of the interior angles of a square is degrees.

The interior angles of a pentagon add up to 540 , so each angle in a regular pentagon is 108 degrees.

60<90<108

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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 inches by inches by inches

b) the volume of a cube with dimensions inches by inches by inches

c) the volume of a cube with dimensions inches by inches by inches

Possible Answers:

(a), (b), and (c) are all equal

(a) is equal to (b) but not (c)

(a) is equal to (c) but not (b)

(a), (b), and (c) are all unequal

Correct answer:

(a) is equal to (b) but not (c)

Explanation:

Find the three volume by multiplying height by length by width:

a) $4\cdot4\cdot4=64$

Half of this volume is .

b) $2\cdot4\cdot4=32$

c) $2\cdot2\cdot2=8$

Remember that we are only looking at half of the volume in a).

Therefore (a) and (b) are equal but (c) is not.

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Example Question #24 : How To Make Geometric Comparisons

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

Possible Answers:

b>a>c

a=b<c

a=b>c

b<a<c

Correct answer:

b<a<c

Explanation:

Area is calculated by multiplying the side lengths:

a) area of a rectangle with side lengths and

$5\cdot3=15$

b) area of a rectangle with side lengths and

$3\cdot4=12$

c) area of a square with side length

$4\cdot4=16$

Therefore (b) is less than (a), which is less than (c).

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Example Question #22 : Geometric Comparison

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

Possible Answers:

$\ a< c< b$

c< b< a

a>c>b

c>b>a

Correct answer:

$\ a< c< b$

Explanation:

To find the side length of a cube from its volume, find the cube root:

$\sqrt[3]{64}=4$

To find the side length of a square from its area, find the square root:

b) $\sqrt{64}=8$

c) $\sqrt{36}=6$

(a) is smaller than (c), which is smaller than (b)

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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 and

Possible Answers:

a>b>c

a<b=c

a=b<c

a=b>c

a=b=c

Correct answer:

a>b>c

Explanation:

First, we find the areas of a, b, and c.

$a=\pi r^2=\pi (5)^2=25\pi$

b=s^2=(5)^2=25

c=(l)(w)=(4)(6)=24

Now we put them in order of size.

a>b>c

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Example Question #22 : 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

Possible Answers:

a=b=c

a>b>c

a<b=c

a=c<b

a<c<b

Correct answer:

a<c<b

Explanation:

First, find the perimeter of the shapes.

Since the area of is , its side length is , giving it a perimeter of .

The perimeter of is $\pi *d=12\pi$ .

The perimeter of is 5*6=30 .

Since $\pi>3$ , $12\pi>36$ .

Therefore, a<c<b .

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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 $\frac{49\pi}{4}$

Possible Answers:

a=b<c

a>b>c

a<b<c

a=b=c

a>b=c

Correct answer:

a=b=c

Explanation:

First, find the lengths given.

Since the square in has area , the side length is . $(A_{square}=side^2)$

All sides of a square have equal lengths, so gives us a length of 28/4=7 .

The area of a circle is $\pi r^2$ , and diameter is . Since area is $\frac{49\pi}{4}$ , $r=\frac{7}{2}$ , giving us d=7 .

The three lengths are equal, so a=b=c .

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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 and height

c. Circle with circumference $24\pi$

Possible Answers:

c<b<a

c=a=b

c=a>b

c>b>a

c>a>b

Correct answer:

c>a>b

Explanation:

First, find the areas.

Since the perimeter of is , its side length is , making the area 121 .

For triangles, $A=\tfrac{1}{2}b*h$ , so the area of is .

For , circumference is $2\pi r$ , making r=12 , giving us an area of $144\pi$ .

Putting them in order, we get c>a>b .

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Example Question #27 : 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

Possible Answers:

b>a=c

a=b=c

a>b>c

a<b<c

a=c>b

Correct answer:

a=c>b

Explanation:

Find the perimeters first.

a=6*(7)=42

b=4*(10)=40

c=3*(14)=42

Putting them in order of size:

a=c>b

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HSPT Quantitative : Geometric Comparison

Study concepts, example questions & explanations for HSPT Quantitative

All HSPT Quantitative Resources

Example Questions

Example Question #21 : Geometric Comparison

Example Question #22 : How To Make Geometric Comparisons

Example Question #23 : How To Make Geometric Comparisons

Example Question #24 : How To Make Geometric Comparisons

Example Question #22 : Geometric Comparison

Example Question #26 : How To Make Geometric Comparisons

Example Question #22 : How To Make Geometric Comparisons

Example Question #27 : How To Make Geometric Comparisons

Example Question #28 : How To Make Geometric Comparisons

Example Question #27 : How To Make Geometric Comparisons

All HSPT Quantitative Resources

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