Geometry - ISEE Upper Level Quantitative Reasoning

Card 0 of 1748

Question

The perimeter of a regular hexagon is one mile. Give the sidelength in feet.

Answer

One mile is equal to 5,280 feet, so divide this by 6:

$5,280 \div 6 = 880$ feet

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Question

If the sum of angles in a hexagon is equal to $650^\circ$ , what is the value of the remaining sixth angle?

Answer

The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.

Given that a hexagon has 6 angles, the total number of angles will be:

$t=4\cdot 180$

Given that 5 of the angles have a sum of 650 degrees, we would subtract 650 from 720, resulting in 70 degrees.

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Question

What is the sidelength, in feet and inches, of a regular hexagon with perimeter 16 feet?

Answer

Divide the perimeter 16 feet by 6 to get the sidelength of the hexagon in feet:

$16 \div 6 = 2$\frac{2}{3}$ \textrm{ ft }$

Two-thirds of a foot is equal to eight inches, so this sidelength is 2 feet 8 inches.

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Question

The perimeter of a hexagon is 96cm. Find the length of one side.

Answer

A hexagon has 6 equal sides. To find the length of one side of a hexagon, we will use the formula to find the perimeter.

where a is the length of one side. So, we will solve for a.

Now, we know the perimeter of the hexagon is 96cm. So, we get

$96$\text{cm}$ = 6a$

$\frac{96\text{cm}$}{6} = $\frac{6a}{6}$

$16$\text{cm}$ = a$

$a = 16$\text{cm}$$

Therefore, the length of one side of the hexagon is 16cm.

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Question

A hexagon has a perimeter of 114in. Find the length of one side.

Answer

A hexagon has 6 equal sides. The formula to find perimeter of a hexagon is:

where a is the length of any side. Now, to find the length of one side, we will solve for a.

We know the perimeter of the hexagon is 114in. So, we will substitute and solve for a. We get

$114$\text{in}$ = 6a$

$\frac{114\text{in}$}{6} = $\frac{6a}{6}$

$19$\text{in}$ = a$

$a = 19$\text{in}$$

Therefore, the length of one side of the hexagon is 19in.

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Question

Find the perimeter of a hexagon with one side having a length of 15cm.

Answer

To find the perimeter of a hexagon, we will use the following formula:

$P = 6 \cdot a$

where a is the length of one side of the hexagon.

Now, we know the length of one side of the hexagon is 15cm.

Knowing this, we can substitute into the formula. We get

$P = 6 \cdot 15$\text{cm}$$

$P = 90$\text{cm}$$

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Question

Find the perimeter of a hexagon with a side having a length of 13cm.

Answer

A hexagon has 6 sides. So, to find the perimeter, we will use the following formula:

where a is the length of one side.

Now, we know a side has a length of 13cm. Knowing this, we can substitute into the formula.

$P = 6 \cdot 13$\text{cm}$$

$P = 78$\text{cm}$$

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Question

A hexagon has a side of length 13cm. Find the perimeter.

Answer

To find the perimeter of a hexagon, we will use the following formula:

where a is the length of one side of the hexagon. We multiply it by 6 because a hexagon has 6 equal sides.

So, we know the length of one side is 13cm. Knowing this, we can substitute into the formula. We get

$P = 6 \cdot 13$\text{in}$$

$P = 78$\text{in}$$

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Question

Find the perimeter of a hexagon with a side having a length of 12in.

Answer

To find the perimeter of a hexagon, we will use the following formula:

$P = 6 \cdot a$

where a is the length of one side of the hexagon. Because a hexagon has 6 equal sides, we can use any side in the formula.

So, we know the length of one side is 12in. So, we can substitute. We get

$P = 6 \cdot 12$\text{in}$$

$P = 72$\text{in}$$

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Question

Find the perimeter of a hexagon with a side having a length of 7cm.

Answer

A hexagon has 6 equal sides. To find the perimeter of a hexagon, we will use the following formula:

where a is the length of one side of the hexagon.

Now, we know one side has a length of 7cm. So, we get

$P = 6 \cdot 7$\text{cm}$$

$P = 42$\text{cm}$$

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Question

Find the perimeter of a hexagon if it contains a side having a length of 17cm.

Answer

To find the perimeter of a hexagon, we will use the following formula:

$P = 6 \cdot a$

where a is the length of one side of the hexagon.

Now, we know the length of one side of the hexagon is 17cm.

Knowing this, we can substitute into the formula. We get

$P = 6 \cdot 17$\text{cm}$$

$P = 102$\text{cm}$$

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Question

Find the volume of a pyramid with the following measurements:

length: 7in

width: 6in

height: 8in

Answer

To find the volume of a pyramid, we will use the following formula:

$V = $\frac{lwh}{3}$$

where l is the length, w is the width_,_ and h is the height of the pyramid.

Now, we know the following measurements:

length: 7in

width: 6in

height: 8in

So, we get

$V = $\frac{ 7\text{in}$ \cdot 6$\text{in}$ \cdot 8$\text{in}$}{3}$

$V = $$\frac{42\text{in}$^2$ \cdot 8$\text{in}$}{3}$

$V = $$\frac{336\text{in}$^3$}{3}$

$V = $112$\text{in}$^3$$

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Question

What is the perimeter of a hexagon with a side length of 1?

Answer

A hexagon has 6 similar sides.

The formula to find the perimeter of a hexagon is:

Substitute the side into the equation.

The perimeter of the hexagon is .

The answer is:

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Question

How many degrees are in an internal angle of a regular heptagon?

Answer

The number of degrees in an internal angle of a regular polygon can be solved using the following equation where n equals the number of sides in the polygon:

$$\frac{180(n-2)}{n}$=\frac{180(7-2)}{7}$=\frac{900}{7}$=128.6$

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Question

What is the measure of an interior angle of a regular nonagon?

Answer

The measure of an interior angle of a regular polygon can be determined using the following equation where n equals the number of sides:

$int=\frac{180(n-2)}{n}$=\frac{180(9-2)}{9}$=\frac{1260}{9}$=140$

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Question

What is the sum of all the interior angles of a decagon (a polygon with ten sides)?

Answer

The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.

$t=8\cdot 180$

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Question

If each angle in a pentagon is equal to , what is the value of $$\frac{1}{2}$x$ ?

Answer

The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.

Given that a hexagon has 6 angles, the total number of angles will be:

$t=3\cdot 180$

To find the value of each angle, we divide 540 by 5. This results in 108 degrees.

Thus,

$$\frac{1}{2}$x=54$

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Question

What is the value of an angle (to the nearest degree) in a polygon with sides if all the angles are equal to one another?

Answer

The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.

Given that a hexagon has 6 angles, the total number of angles will be:

$t=20\cdot 180$

Given that there are 3,600 degrees total in a polygon with 22 sides, the number of degrees in each angle can be found by dividing 3,600 by 22. To the nearest degree, this results in 164 degrees. Therefore, 164 is the correct answer.

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Question

What is $\frac{1}{7}$ of the total number of degrees in a 9-sided polygon?

Answer

The sum of the angles in a polygon can be found using the equation below, in which t is equal to the total sum of the angles, and n is equal to the number of sides.

Therefore, the equation for the sum of the angles in a 9 sided polygon would be:

$t=7\cdot 180$

Therefore, $\frac{1}{7}$ of the total sum of degrees in a 9 sided polygon would be equal to 180 degrees.

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Pentagon is regular. What is the measure of $\angle 1$ ?

Answer

The answer can be more clearly seen by extending $\overline{PA}$ to a ray $\overrightarrow{PA}$ :

Note that angles have been newly numbered.

$\angle 2$ and $\angle 4$ are exterior angles of a (five-sided) regular pentagon in relation to two parallel lines, so each has a measure of $\left (360 \div 5 \right $)^{\circ}$= $72^{\circ}$$ . $\angle 3$ is a corresponding angle to $\angle 4$ , so its measure is also .

By angle addition,

$m \angle 1 = m\angle 2 + m\angle 3 = $72^{\circ}$+ $72^{\circ}$ = $144^{\circ}$$

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