# HSPT Quantitative : How to make geometric comparisons

## Example Questions

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### Example Question #1 : Hspt Quantitative Skills

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

Possible Answers:

Correct answer:

Explanation:

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 #1 : How To Make Geometric Comparisons

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

Possible Answers:

Correct answer:

Explanation:

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 #3 : Hspt Quantitative Skills

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

Possible Answers:

Correct answer:

Explanation:

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).

### Example Question #2 : How To Make Geometric Comparisons

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

Possible Answers:

(a) and (b) are equal

(c) is greater than (a) and (b)

(b) and (c) are equal

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

Correct answer:

(b) and (c) are equal

Explanation:

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).

### Example Question #3 : How To Make Geometric Comparisons

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

Possible Answers:

(a) and (c) are equal

(b) is greater than (a) or (c)

(b) and (c) are equal

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

Correct answer:

(b) and (c) are equal

Explanation:

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.

### Example Question #6 : Hspt Quantitative Skills

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

Possible Answers:

(c) is greater than (a) and (b).

(a) and (c) are equal.

(a) and (b) are equal.

(a) is greater than (b) and (c).

Correct answer:

(c) is greater than (a) and (b).

Explanation:

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.

### Example Question #4 : How To Make Geometric Comparisons

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

Possible Answers:

Correct answer:

Explanation:

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 : Hspt Quantitative Skills

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

Possible Answers:

(a) has the greatest area.

(b) has the greatest area.

(c) has the greatest area.

(a), (b), and (c) have equal area.

Correct answer:

(c) has the greatest area.

Explanation:

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.

### Example Question #5 : 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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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).

### Example Question #9 : Hspt Quantitative Skills

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

Possible Answers:

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

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

(a), (b), and (c) are all not equivalent

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

Correct answer:

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

Explanation:

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.

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