### All HSPT Quantitative Resources

## 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 = b < c

c < b < a

a = c < b

**Correct answer:**

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 .

(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

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

(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

**Correct answer:**

(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 #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

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

**Correct answer:**

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?

a. A circle with radius

b. A square with side

c. A rectangle with side lengths of and

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

**Correct answer:**

Find the perimeters first.

Putting them in order of size:

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