Pentagons - Math

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Question

A pentagon with a perimeter of one mile has three congruent sides. The second-longest side is 250 feet longer than any one of those three congruent sides, and the longest side is 500 feet longer than the second-longest side.

Which is the greater quantity?

(a) The length of the longest side of the pentagon

(b) Twice the length of one of the three shortest sides of the pentagon

Answer

If each of the five congruent sides has measure , then the other two sides have measures and $\left (x + 250 \right ) +500 = x + 750$ . Add the sides to get the perimeter, which is equal to feet, the solve for :

$5x\div 5 = 4,280 \div 5$

feet

Now we can compare (a) and (b).

(a) The longest side has measure feet.

(b) The three shortest sides each have length 856 feet; twice this is $856 \times 2 = 1,712$ feet.

(b) is greater.

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Question

A regular pentagon has perimeter one yard. Which is the greater quantity?

(A) The length of one side

(B) 7 inches

Answer

One yard is equal to 36 inches. A regular pentagon has five sides of equal length, so one side of the pentagon has length

$36 \div 5 = 7 $\frac{1}{5}$$ inches.

Since $7 $\frac{1}{5}$ > 7$ , (A) is greater.

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Question

A regular pentagon has a side length of inches and an apothem length of inches. Find the area of the pentagon.

Answer

By definition a regular pentagon must have equal sides and equivalent interior angles. Since we are told that this pentagon has a side length of inches, all of the sides must have a length of inches. Additionally, the question provides the length of the apothem of the pentagon--which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$$

$A=\frac{9\times 6.2}{2}$=\frac{55.8}{2}$=27.9$

Note: the area shown above is only the a measurement from one of the five total interior triangles. Thus, to find the total area of the pentagon multiply:

$27.9\times5=139.5$

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Question

A regular pentagon has an area of $\small 32.5$ square units and an apothem measurement of $\small \small 3$ . Find the length of one side of the pentagon.

Answer

A regular pentagon must have five equivalent sides and five equivalent interior angles. Regular pentagons can be divided up into five equivalent interior triangles, where the base of the triangle is a side length and the height of the triangle is the apothem. This problem provides the total area for the pentagon.

Therefore, you must work backwards using the area formula: $\small a=\frac{base\times height}{2}$$ , where the base is equal to the length of one side of the triangle and the height of the triangle is the measurement of the apothem. However, this will only provide the measurement for one of the five interior triangles, thus you first need to divide the total area by $\small 5.$

The solution is:

$\small \small a=32.5$

So, area of one of the five interior triangles is equal to: $\small \small 32.5\div5=6.5$

Now, apply the area formula:

$\small \small 6.5=\frac{base\times 3}{2}$$

$\small \small 13=base\times3$

$\small \small base=13\div3=4.3$

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Question

A regular pentagon has an area of $\small \small 156$ square units and an apothem measurement of $\small \small 6.6$ . Find the length of one side of the pentagon.

Answer

A regular pentagon must have five equivalent sides and five equivalent interior angles. Regular pentagons can be divided up into five equivalent interior triangles, where the base of the triangle is a side length and the height of the triangle is the apothem. This problem provides the total area for the pentagon.

Therefore, you must work backwards using the area formula: $\small a=\frac{base\times height}{2}$$ , where the base is equal to the length of one side of the triangle and the height of the triangle is the measurement of the apothem. However, this will only provide the measurement for one of the five interior triangles, thus you first need to divide the total area by $\small 5.$

The solution is:

$\small \small a=156$

So, area of one of the five interior triangles is equal to: $\small \small 156\div5=31.2$

Now, apply the area formula:

$\small \small 31.2=\frac{base\times 6.6}{2}$$

$\small \small 62.4=base\times6.6$

$\small \small base=62.4\div 6.6= 9.5$

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Question

Find the area of the regular pentagon.

Answer

Recall that the area of regular polygons can be found using the following formula:

$$\text{Area of Regular Polygon}$=\frac{1}{2}$($\text{apothem}$)($\text{perimeter}$)$

First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:

$$\text{Perimeter}$=5($\text{side length}$)$

Substitute in the length of the given side of the pentagon in order to find the perimeter.

$$\text{Perimeter}$=5(7)$

$$\text{Perimeter}$=35$

Next, substitute in the given and calculated information to find the area of the pentagon.

$$\text{Area of Pentagon}$=\frac{1}{2}$(4)(35)$

$$\text{Area of Pentagon}$=70$

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Question

The angles at 3 verticies of a pentagon are 60, 80 and 100. Which of the following could NOT be the measures of the other 2 angles?

Answer

The sum of the angles in a polygon is

For a pentagon, this equals 540. Since the first 3 angles add up to 240, the remaining 2 angles must add up to

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Question

What is the sum of three angles in a hexagon if the perimeter of the hexagon is ?

Answer

The perimeter in this question is irrelevant. Use the interior angle formula to determine the total sum of the angles in a hexagon.

$\Theta =(n-2)\cdot180$

There are six interior angles in a hexagon.

$\Theta =(6-2)\cdot180 = $720^{\circ}$$

Each angle will be a sixth of the total angle.

$\frac{720}{6}$= $120^{\circ}$

Therefore, the sum of three angles in a hexagon is:

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Question

Add four interior angles in a regular pentagon. What is the result?

Answer

Use the interior angle formula to find the total sum of angles in a pentagon.

$\sum \Theta=(n-2) \cdot 180$

for a pentagon, so substitute this value into the equation and solve:

$\sum \Theta=(5-2) \cdot 180 = $540^{\circ}$$

Divide this number by 5, since there are five interior angles.

The sum of four interior angles in a regular pentagon is:

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Question

What is the sum of two interior angles of a regular pentagon if the perimeter is 6?

Answer

The perimeter of a regular pentagon has no effect on the interior angles of the pentagon.

Use the following formula to solve for the sum of all interior angles in the pentagon.

$\sum\theta=(n-2)\cdot 180$

Since there are 5 sides in a pentagon, substitute the side length .

$\sum\theta=(5-2)\cdot 180=540$

Divide this by 5 to determine the value of each angle, and then multiply by 2 to determine the sum of 2 interior angles.

$$\frac{540}{5}$=108$

The sum of 2 interior angles of a pentagon is .

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Question

Let the area of a regular pentagon be . What is the value of an interior angle?

Answer

Area has no effect on the value of the interior angles of a pentagon. To find the sum of all angles of a pentagon, use the following formula, where is the number of sides:

$\sum\theta=(n-2)\cdot 180$

There are 5 sides in a pentagon.

$\sum\theta=(5-2)\cdot 180=540$

Divide this number by 5 to determine the value of each angle.

$$\frac{540}{5}$=108^\circ$

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Question

Suppose an interior angle of a regular pentagon is . What is ?

Answer

The pentagon has 5 sides. To find the value of the interior angle of a pentagon, use the following formula to find the sum of all interior angles.

$\sum\theta=(n-2)\cdot 180$

Substitute .

$\sum\theta=(5-2)\cdot 180=540$

Divide this number by 5 to determine the value of each interior angle.

$$\frac{540}{5}$=108$

Every interior angle is 108 degrees. The problem states that an interior angle is . Set these two values equal to each other and solve for .

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Question

True or false: Each of the five angles of a regular pentagon measures .

Answer

A regular polygon with sides has congruent angles, each of which measures

Setting , the common angle measure can be calculated to be

The statement is therefore false.

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Question

True or false: Each of the exterior angles of a regular pentagon measures $72 ^{\circ }$ .

Answer

If one exterior angle is taken at each vertex of any polygon, and their measures are added, the sum is . Each exterior angle of a regular pentagon has the same measure, so if we let be that common measure, then

$5t = 360 ^{\circ }$

Solve for :

$5t \div 5 = 360 ^{\circ } \div 5$

$t = 72 ^{\circ }$

The statement is true.

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Question

Given: Pentagon .

$m \angle P = 108 ^{\circ }$

True, false, or undetermined: Pentagon is regular.

Answer

Suppose Pentagon is regular. Each angle of a regular polygon of sides has measure

A pentagon has 5 sides, so set ; each angle of the regular hexagon has measure

Since one angle is given to be of measure $108 ^{\circ }$ , the pentagon might be regular - but without knowing more, it cannot be determined for certain. Therefore, the correct choice is "undetermined".

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Question

A regular pentagon has a side length of and an apothem length of . Find the area of the pentagon.

Answer

By definition a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon--which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$$

$A=\frac{14\times 9.6}{2}$=\frac{134.4}{2}$=67.2$

Note: is only the measurement for one of the five interior triangles. Thus, the final solution is:

$67.2\times5=336$

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Question

Find the area of the pentagon shown above.

Answer

To find the area of this pentagon, divide the interior of the pentagon into a four-sided rectangle and two right triangles. The area of the bottom rectangle can be found using the formula:

$A=width\times length$

$A=20\times15=300$

The area of the two right triangles can be found using the formula:

$A=\frac{base\times height}{2}$$

$A=\frac{10\times 8}{2}$=\frac{80}{2}$=40$

Since there are two right triangles, the sum of both will equal the area of the entire triangular top portion of the pentagon.

Thus, the solution is:

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Question

A regular pentagon has a perimeter of yards and an apothem length of yards. Find the area of the pentagon.

Answer

To solve this problem, first work backwards using the perimeter formula for a regular pentagon:

$s=\frac{60}{5}$=12$

Now you have enough information to find the area of this regular triangle.
Note: a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$$

$A=\frac{8.3\times 12}{2}$=\frac{99.6}{2}$=49.8$

Thus, the area of the entire pentagon is:

$A=49.8\times 5=249$

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Question

A regular pentagon has a side length of and an apothem length of . Find the area of the pentagon.

Answer

By definition a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$$

$A=\frac{15\times 10.3}{2}$=\frac{154.5}{2}$=77.25$

Keep in mind that this is the area for only one of the five total interior triangles.

The total area of the pentagon is:

$A=77.25\times 5=386.25$

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Question

A regular pentagon has a perimeter of and an apothem length of . Find the area of the pentagon.

Answer

To solve this problem, first work backwards using the perimeter formula for a regular pentagon:

$s=\frac{50}{5}$=10$

Now you have enough information to find the area of this regular triangle.
Note: a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of and a height of .

The area of this pentagon can be found by applying the area of a triangle formula:

$A=\frac{base\times height}{2}$$

$A=\frac{10\times 7}{2}$=\frac{70}{2}$=35$

To find the total area of the pentagon multiply:

$35\times5=175$

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