How to find the length of a side

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ISEE Upper Level Quantitative Reasoning › How to find the length of a side

Questions 1 - 5

A regular pentagon has sidelength 72; the perimeter of a regular hexagon is 80% of that of the pentagon. Which is the greater quantity?

(A) The length of one side of the hexagon

(B) 50

(B) is greater

(A) is greater

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

(A) and (B) are equal

Explanation

A regular pentagon has five sides of equal length; since one side is 72 units long, its perimeter is

$72 \times 5 = 360$ .

80% of this is

$0.8 \times 360 = 288$ ,

so this is the length of the hexagon, and, since all six sides are of equal length, one side measures

$288 \div 6 = 48$

(B) is greater.

A regular octagon has perimeter one meter. Which is the greater quantity?

(A) The length of one side

(B) 125 millimeters

(A) and (B) are equal

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

(A) is greater

(B) is greater

Explanation

A regular octagon has eight sides of equal length. The perimeter of this octagon is one meter, which is equal to 1,000 millimeters; each side, therefore, has length

$1,000 \div 8 = 125$ millimeters

making the quantities equal.

Right_triangle

A regular decagon has the same perimeter as the above right triangle. Give the length of one side.

$11 \frac{1}{5} \textrm{ in}$

$14 \textrm{ in}$

$9 \frac{1}{3} \textrm{ in}$

$6 \frac{1}{5 }\textrm{ in}$

$7 \frac{3}{4} \textrm{ in}$

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

inches.

A regular decagon has ten sides of equal length, so each side measures

$112 \div 10 = 11 \frac{1}{5}$ inches.

A regular octagon has twice the perimeter of a regular pentagon. What is the ratio of the sidelength of the octagon to that of the pentagon?

$5\textrm{ to }4$

$4\textrm{ to } 5$

$5\textrm{ to }8$

$8\textrm{ to }5$

$5\textrm{ to }2$

Explanation

The solution is independent of the actual lengths, so we assume the pentagon has sidelength 1. Its perimeter is therefore 5. Subsequently, the octagon's perimeter is twice this, or 10, and its sidelength is one-eighth of this, or

$\frac{10}{8} = \frac{5}{4}$ .

The ratio of the sidelength of the octagon to that of the pentagon is

$\frac{5}{4} : 1$ or 5 to 4.

A regular octagon has perimeter one mile.Which is the greater quantity?

(a) The length of one side

(b) 880 feet

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

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

Explanation

A regular octagon has eight sides of equal length. The perimeter, which is the sum of the lengths of these sides, is one mile, which is equal to 5,280 feet. Therefore, the length of one side is

$5, 280 \textrm{ ft} \div 8 = 660 \textrm{ ft}$ . This makes the length of a side less than 880 feet.

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