# HSPT Quantitative : How to make geometric comparisons

## 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

c > b > a

a = b < c

c < b < a

a = c < b

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:

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).

### Example Question #22 : Geometric Comparison

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

Explanation:

Since the interior angles of a triangle add up to , 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 , so each angle in a regular pentagon is  degrees.

### Example Question #23 : Geometric Comparison

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

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

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

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

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

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

Explanation:

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 #21 : Geometric Comparison

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

Explanation:

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).

### Example Question #21 : 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  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

Explanation:

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 : Geometric Comparison

What are the relationships between the areas of these shapes?

b. A square with side

c. A rectangle with side lengths of and

Explanation:

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

Now we put them in order of size.

### Example Question #27 : Geometric Comparison

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

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 .

The perimeter of is .

Since , .

Therefore, .

### Example Question #28 : Geometric Comparison

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

Explanation:

First, find the lengths given.

Since the square in  has area , the side length is

All sides of a square have equal lengths, so gives us a length of .

The area of a circle is , and diameter is . Since area is , , giving us .

The three lengths are equal, so .

### Example Question #29 : Geometric Comparison

Find the relationship between the areas of the following shapes.

a. Square with perimeter

b. Triangle with base and height

c. Circle with circumference

Explanation:

First, find the areas.

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

For triangles, , so the area of is .

For , circumference is , making , giving us an area of .

Putting them in order, we get .

### Example Question #30 : Geometric Comparison

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