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ISEE Upper Level Quantitative : Hexagons

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

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Hexagons

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:

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

Correct answer:

(a) is greater

Explanation:

A regular hexagon with sidelength s = 1 can be seen as a composite of six equilateral triangles, each with sidelength s = 1 . Since area is in direct proportion to the square of the sidelength, the area of the equilateral triangle with sidelength s = 2 is equal to that of four of those triangles. This makes the hexagon greater in area, and it makes (a) the greater quantity.

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Example Question #1 : 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:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

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, $5 \times 12 = 60$ inches.

(b) Multiply: $6 \times 10 = 60$ inches

The two polygons have the same perimeter.

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Example Question #94 : Geometry

A hexagon has six angles with measures $(x-5)^{\circ}, x^{\circ}, (x+5)^{\circ}, (y-10)^{\circ}, y^{\circ}, (y + 10)^{\circ}.$

Which quantity is greater?

(a) x + y

(b) 240

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) and (b) are equal

Explanation:

The angles of a hexagon measure a total of 180 (6-2) = 720 . From the information, we know that:

(x - 5) + x + (x+5) + (y-10) + y + (y+10) = 720

x+ x + x + y+ y + y - 5+5 -10 +10= 720

3x + 3y = 720

3 (x + y) = 720

$3 (x + y) \div 3 = 720 \div 3$

x + y = 240

The quantities are equal.

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Example Question #1 : Hexagons

A hexagon has six angles with measures $(x-10)^{\circ}, x^{\circ}, (x+5)^{\circ}, (y-10)^{\circ}, y^{\circ}, (y + 20)^{\circ}.$

Which quantity is greater?

(a) x + y

(b) 240

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(b) is greater.

Explanation:

The angles of a hexagon measure a total of 180 (6-2) = 720 . From the information, we know that:

(x-10)+x+(x+5)+ (y-10)+ y+ (y + 20)= 720

x + x + x + y + y + y -10+5 -10 + 20= 720

3x + 3y +5= 720

3x + 3y +5-5= 720-5

3x + 3y = 715

$3\left ( x + y \right )= 715$

$3\left ( x + y \right ) \div 3= 715\div 3$

$x + y = 238 \frac{1}{3} < 240$

This makes (b) greater.

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Example Question #96 : Geometry

The angles of Hexagon A measure

$A ^{\circ }, B ^{\circ }, 150 ^{\circ }, 150 ^{\circ }, 150 ^{\circ }, 150 ^{\circ }$

The angles of Octagon B measure

$C ^{\circ }, D^{\circ }, 150^{\circ },150^{\circ },150^{\circ },150^{\circ },150^{\circ },150^{\circ }$

Which is the greater quantity?

(A) A + B

(B) C + D

Possible Answers:

(A) is greater

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

(B) is greater

(A) and (B) are equal

Correct answer:

(B) is greater

Explanation:

The sum of the measures of a hexagon is $180 ^{\circ } \times (6-2)= 720 ^{\circ }$ . Therefore,

A + B + 150 + 150 + 150 + 150 = 720

A + B +600= 720

A + B = 120

The sum of the measures of an octagon is $180 ^{\circ } \times (8-2)= 1,080 ^{\circ }$ . Therefore,

C + D + 150 + 150 + 150 + 150+ 150 + 150 = 1,080

C + D + 900 = 1,080

C + D = 180

C + D > A + B , so (B) is greater.

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Example Question #97 : Geometry

The angles of Pentagon A measure $110^{\circ }, 110^{\circ }, 110^{\circ }, y^{\circ }, y^{\circ }$

The angles of Hexagon B measure $130^{\circ }, 130^{\circ }, 130^{\circ }, 130^{\circ },z^{\circ }, z^{\circ }$

Which is the greater quantity?

(A)

(B)

Possible Answers:

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

(B) is greater

(A) is greater

(A) and (B) are equal

Correct answer:

(A) is greater

Explanation:

The sum of the measures of the angles of a pentagon is $180^{\circ } \times (5-2) =540^{\circ }$ . Therefore,

y + y + 110 + 110 + 110 = 540

2y +330 = 540

2y = 210

y = 105

The sum of the measures of a hexagon is $180^{\circ } \times (6-2) =720 ^{\circ }$ . Therefore,

z + z + 130 + 130 + 130 + 130 = 720

2z + 520 = 720

2z= 200

z = 100

y> z , so (A) is greater.

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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.

$22 \frac{2}{5} \textrm{ in}$

$10 \frac{1}{3}\textrm{ in}$

$12 \frac{2}{5}\textrm{ in}$

$18 \frac{2}{3} \textrm{ in}$

Correct answer:

$18 \frac{2}{3} \textrm{ in}$

Explanation:

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

$\sqrt{14^{2}+48^{2}} = \sqrt{196+2,304} = \sqrt{2,500} = 50$ inches, making its perimeter

14 + 48 + 50 =112 inches.

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

$112 \div 6 = 18 \frac{2}{3}$ inches.

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Example Question #2 : 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:

$22 \frac{2}{5} \textrm{ in}$

The length cannot be determined from the information given.

$10 \frac{1}{3}\textrm{ in}$

$18 \frac{2}{3} \textrm{ in}$

$12 \frac{2}{5}\textrm{ in}$

Correct answer:

$18 \frac{2}{3} \textrm{ in}$

Explanation:

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

$\sqrt{14^{2}+48^{2}} = \sqrt{196+2,304} = \sqrt{2,500} = 50$ inches, making its perimeter

14 + 48 + 50 =112 inches.

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

$112 \div 6 = 18 \frac{2}{3}$ inches.

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Hexagons

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : Hexagons

Example Question #1 : Hexagons

Example Question #94 : Geometry

Example Question #1 : Hexagons

Example Question #96 : Geometry

Example Question #97 : Geometry

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

Example Question #2 : How To Find The Length Of The Side Of A Hexagon

All ISEE Upper Level Quantitative Resources

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