ISEE Upper Level Quantitative : Hexagons

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

Example Question #91 : Geometry

Which is the greater quantity?

(a) The area of a regular hexagon with sidelength 1

(b) The area of an equilateral triangle with sidelength 2

Possible Answers:

(a) is greater

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(a) is greater

Explanation:

A regular hexagon with sidelength  can be seen as a composite of six equilateral triangles, each with sidelength . Since area is in direct proportion to the square of the sidelength, the area of the equilateral triangle with sidelength  is equal to that of four of those triangles. This makes the hexagon greater in area, and it makes (a) the greater quantity.

Example Question #2 : Hexagons

Which is the greater quantity?

(a) The perimeter of a regular pentagon with sidelength 1 foot

(b) The perimeter of a regular hexagon with sidelength 10 inches

Possible Answers:

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

The sides of a regular polygon are congruent, so in each case, multiply the sidelength by the number of sides to get the perimeter.

(a) Since one foot equals twelve inches,  inches.

(b) Multiply:  inches

The two polygons have the same perimeter.

Example Question #1 : Hexagons

A hexagon has six angles with measures 

Which quantity is greater?

(a) 

(b) 240

Possible Answers:

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

The angles of a hexagon measure a total of . From the information, we know that:

The quantities are equal.

Example Question #4 : Hexagons

A hexagon has six angles with measures 

Which quantity is greater?

(a) 

(b)

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

Correct answer:

(b) is greater.

Explanation:

The angles of a hexagon measure a total of .  From the information, we know that:

This makes (b) greater.

Example Question #5 : Hexagons

The angles of Hexagon A measure 

The angles of Octagon B measure 

Which is the greater quantity?

(A) 

(B) 

Possible Answers:

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(B) is greater

(A) is greater

Correct answer:

(B) is greater

Explanation:

The sum of the measures of a hexagon is  . Therefore,

 

The sum of the measures of an octagon is . Therefore,

 

, so (B) is greater.

Example Question #6 : Hexagons

The angles of Pentagon A measure 

The angles of Hexagon B measure 

Which is the greater quantity?

(A) 

(B) 

Possible Answers:

(B) is greater

(A) and (B) are equal

(A) is greater

It is impossible to determine which is greater from the information given

Correct answer:

(A) is greater

Explanation:

The sum of the measures of the angles of a pentagon is . Therefore, 

 

The sum of the measures of a hexagon is  . Therefore,

 

, so (A) is greater.

Example Question #1 : How To Find The Length Of The Side Of A Hexagon

Right_triangle

A regular hexagon has the same perimeter as the above right triangle. What is the length of one side of the hexagon?

Possible Answers:

The length cannot be determined from the information given.

Correct answer:

Explanation:

By the Pythagorean Theorem, the hypotenuse of the right triangle is 

 inches, making its perimeter

 inches.

The regular hexagon, which has six sides of equal length, has the same perimeter, so each side measures

 inches.

Example Question #101 : Geometry

Right_triangle

A regular hexagon has the same perimeter as the above right triangle. What is the length of one side of the hexagon?

Possible Answers:

The length cannot be determined from the information given.

Correct answer:

Explanation:

By the Pythagorean Theorem, the hypotenuse of the right triangle is 

 inches, making its perimeter

 inches.

The regular hexagon, which has six sides of equal length, has the same perimeter, so each side measures

 inches.

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