Find the values of a, b, and c.

Since is closest to , the answer is c.

Find the perimeters of the shapes.

A rhombus has four identical sides:

A hexagon has six:

A pentagon has five:

The correct relationship is .

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

 is represented by four colored portions out of eight, so .

 is represented by six colored portions out of twelve, so 

 is represented by four colored portions out of six, so .

. Since

and 

,

it follows that .

Circle  has radius 4; its diameter is twice this, which is 8.

Circle  has area . Since the area is equal to  times the square of the radius, that is, 

,

substitute  for the area and solve for :

The diameter of the circle is twice this, or 4.

Circle  has circumference ; its diameter is the circumference divided by , which is

Therefore, .

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

Circle B has area , so substitute this for the area:

, 

the radius of circle B.

The radius of a circle is its circumference divided by . Circle C has circumference , so divide this by :

 and , so .

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

Note that each square has been divided into eight congruent parts. , , and  are represented by 3, 4, and 5 colored portions, respectively, so

,

and .

Of the four choices, only  is correct.

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."

Figure A is divided into twelve regions of equal area; nine are shaded in, which make up  of the area. Reduce this to lowest terms by dividing both halves by greatest common factor 3:

.

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

Figure B has the same shaded area as Figure A.

Figure C is divided into eight 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 4:

.

Figure C has the same shaded area as Figure A.

The correct response is "Both Figure B and Figure C."

Below are the same figures, but with some additional lines drawn so that each square is divided into equal parts.

 is represented by 6 parts out of 12, so .

 is represented by 4 parts out of 9, so .

 is represented by 3 parts out of 8, so .

These fractions can be compared by expressing them in terms of a least common denominator - :

Comparing numerators, we see that .

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)