Geometric Comparison

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HSPT Quantitative › Geometric Comparison

Questions 1 - 10
1

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 is equivalent to Circle but not

Circle is equivalent to Circle but not

Circles , , and are all equivalent

Circles , , and are all nonequivalent

Explanation

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 .

2

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

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

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

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

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

Explanation

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.

3

Find the relationship between the areas of the following shapes.

a. A square of side length

b. A parallelogram of base and height

c. A triangle with base and height

Explanation

First, find the areas.

The correct choice is .

4

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, .

5

Find the relationship between the areas of the following shapes.

a. A square of side length

b. A parallelogram of base and height

c. A triangle with base and height

Explanation

First, find the areas.

The correct choice is .

6

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 is equivalent to Circle but not

Circle is equivalent to Circle but not

Circles , , and are all equivalent

Circles , , and are all nonequivalent

Explanation

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 .

7

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

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

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

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

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

Explanation

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.

8

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, .

9

Figuires

Examine the above three figures. All three squares have the same area.

Of Figure B and Figure C, which figure(s) have the same area shaded in as does Figure A?

C but not B

B but not C

Both B and C

Neither B nor C

Explanation

Figure A is divided into twelve regions of equal area; six are shaded in, which make up of the area. Reduce this to lowest terms by dividing both halves by greatest common factor 6:

.

Figure B is divided into sixteen regions of equal area; seven are shaded in, which make up of the area. This is in lowest terms, since 7 and 12 are relatively prime. , so Figure B does not have the same shaded area as Figure A.

Figure C is divided into eight regions of equal area; four are shaded in, which make up of the area. Reduce this to lowest terms by dividing both halves by greatest common factor 4:

.

Figure C has the same shaded area as Figure A.

The correct response is "Figure C, but not Figure B."

10

Figuires

Examine the above three figures. All three squares have the same area.

Of Figure B and Figure C, which figure(s) have the same area shaded in as does Figure A?

C but not B

B but not C

Both B and C

Neither B nor C

Explanation

Figure A is divided into twelve regions of equal area; six are shaded in, which make up of the area. Reduce this to lowest terms by dividing both halves by greatest common factor 6:

.

Figure B is divided into sixteen regions of equal area; seven are shaded in, which make up of the area. This is in lowest terms, since 7 and 12 are relatively prime. , so Figure B does not have the same shaded area as Figure A.

Figure C is divided into eight regions of equal area; four are shaded in, which make up of the area. Reduce this to lowest terms by dividing both halves by greatest common factor 4:

.

Figure C has the same shaded area as Figure A.

The correct response is "Figure C, but not Figure B."

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