HSPT Quantitative - How to make geometric comparisons

Find the relationship between the areas of the following shapes.

a. A square of side length

b. A parallelogram of base and height $\frac{1}{2}$

c. A triangle with base and height

Explanation

First, find the areas.

$b=(b)(h)=(2)*(\tfrac{1}{2})=1$

$c=\tfrac{1}{2}(b)(h)=(\tfrac{1}{2})(1)(2)=1$

The correct choice is .

Examine (a), (b), and (c) to find the best answer:

a) area of a circle with a circumference of $6\pi$

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 $\pi$ . Since the circumfrence in (a) is the diameter in (c) multiplied by $\pi$ , this is also equal.

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 $\pi *d=12\pi$ .

The perimeter of is .

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

Therefore, .

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 $\frac{1}{2}x$

b) Circle has a circumference of $\frac{1}{2}y$

c) Circle has an area of $\frac{1}{2}z$

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 ( $C=2\pi r$ ). 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 ( $A=\pi r^2$ ). 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 .

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 $\frac{6}{12}$ of the area. Reduce this to lowest terms by dividing both halves by greatest common factor 6:

$\frac{6}{12} =\frac{6 \div 6}{12\div 6 } = \frac{1}{2}$ .

Figure B is divided into sixteen regions of equal area; seven are shaded in, which make up $\frac{7}{16}$ of the area. This is in lowest terms, since 7 and 12 are relatively prime. $\frac{7}{16} e \frac{1}{2 }$ , 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 $\frac{4}{8}$ of the area. Reduce this to lowest terms by dividing both halves by greatest common factor 4:

$\frac{4 \div 4}{8 \div 4} = \frac{1}{2}$ .

Figure C has the same shaded area as Figure A.

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

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

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.

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

Explanation

Find the perimeters first.

Putting them in order of size:

Comparison

, , and each refer to the fractions of their respective figures that are gray.

Which statement is correct?

Explanation

Examine the diagram below, which shows all three figures with additional lines.

Comparison

is represented by four colored portions out of eight, so $A = \frac{4}{8} = \frac{1}{2}$ .

is represented by six colored portions out of twelve, so $B = \frac{6}{12} = \frac{1}{2}$

is represented by four colored portions out of six, so $C = \frac{4}{6} = \frac{2}{3}$ .

. Since

$C = \frac{2}{3} = \frac{2 \cdot 2 }{3 \cdot 2} = \frac{4}{6}$

and

$A = B = \frac{1}{2} = \frac{1 \cdot 3 }{2 \cdot 3} = \frac{3}{6}$ ,

it follows that .

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

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

(a) and (b) are equal.

(a) and (c) are equal.

Explanation

Things to remember here are that area is found by squaring side length and that side length is $\frac{1}{4}$ of the perimeter.

$sl=8\div4=2$

Comparison

Note: Figures NOT drawn to scale.

In the above diagram, let , , and represent the radii of their respective circles.

Which is a correct statement?

Explanation

The radius of a circle is one half its diameter. Circle A has diameter 16, so its radius is half this, or 8.

The relationship between the area of a circle and its radius is given by the equation

$\pi r^{2} = Area$

Circle B has area $16 \pi$ , so substitute this for the area:

$\pi r^{2} = 16 \pi$

$\frac{ \pi r^{2}}{\pi} = \frac{16 \pi}{\pi}$

$r^{2} =16$

$r = \sqrt{16} = 4$ ,

the radius of circle B.

The radius of a circle is its circumference divided by $2 \pi$ . Circle C has circumference $16 \pi$ , so divide this by $2 \pi$ :

$\frac{16 \pi}{2 \pi} = 8$

and , so .

How to make geometric comparisons

Help Questions

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation