Pentagons

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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

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Explanation

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.

Each side of this pentagon has a length of .

Solve for the area of the pentagon.

Varsity record

Explanation

The formula for area of a pentagon is $A=5\cdot(\frac{1}{2}\cdot s\cdot h)$ , with representing the length of one side and representing the apothem.

To find the apothem, we can convert our one pentagon into five triangles and solve for the height of the triangle:

Varsity record

Each of these triangles have angle measures of $72$^{\circ}$; 54$^{\circ}$;54$^{\circ}$$ , with $72$^{\circ}$$ being the angle oriented around the vertex. This is because the polygon has been divided into five triangles and $\frac{360}{5}= 72$ .

Varsity pentagon

To solve for the apothem, we can use basic trigonometric ratios:

$tan(54$^{\circ}$)= 1.3764$

$\1.3764=\frac{x}{2.5;cm}\x=3.44;cm$

Now that we know the apothem length, we can plug in all our values to solve for area:

$Area=5\cdot\frac{1}{2}\cdot 5;cm \cdot3.44;cm=43;cm$

If a regular pentagon has an area of and an apothem length of , what is the length of a side of the pentagon?

Explanation

Recall how to find the area of a regular pentagon:

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

Now, the perimeter of a regular pentagon can be found by multiplying the side length by :

$\text{perimeter}=5(\text{side})$

Substitute this into the equation for the area.

$\text{Area}=\frac{1}{2}(\text{apothem})(5(\text{side}))$

Now, rearrange the equation to solve for the side length.

$5(\text{apothem})(\text{side})=2(\text{Area})$

$\text{side}=\frac{2(\text{Area})}{5(\text{apothem})}$

Plug in the given area and apothem to solve for the side length of the pentagon.

$\text{side}=\frac{2(50)}{5(10)}=2$

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

sq. units

Explanation

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

$s=\frac{145}{5}=29$

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:

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

$A=\frac{29\times 20}{2}=\frac{580}{2}=290$

Thus, the area of the entire pentagon is:

$A=290\times 5=1,450$

If a regular pentagon has an area of and an apothem length of , what is the length of a side of the pentagon?

Explanation

Recall how to find the area of a regular pentagon:

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

Now, the perimeter of a regular pentagon can be found by multiplying the side length by :

$\text{perimeter}=5(\text{side})$

Substitute this into the equation for the area.

$\text{Area}=\frac{1}{2}(\text{apothem})(5(\text{side}))$

Now, rearrange the equation to solve for the side length.

$5(\text{apothem})(\text{side})=2(\text{Area})$

$\text{side}=\frac{2(\text{Area})}{5(\text{apothem})}$

Plug in the given area and apothem to solve for the side length of the pentagon.

$\text{side}=\frac{2(50)}{5(10)}=2$

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

sq. units

Explanation

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

$s=\frac{145}{5}=29$

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.

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

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

$A=\frac{29\times 20}{2}=\frac{580}{2}=290$

Thus, the area of the entire pentagon is:

$A=290\times 5=1,450$

Each side of this pentagon has a length of .

Solve for the area of the pentagon.

Varsity record

Explanation

The formula for area of a pentagon is $A=5\cdot(\frac{1}{2}\cdot s\cdot h)$ , with representing the length of one side and representing the apothem.

To find the apothem, we can convert our one pentagon into five triangles and solve for the height of the triangle:

Varsity record

Varsity pentagon

To solve for the apothem, we can use basic trigonometric ratios:

$tan(54$^{\circ}$)= 1.3764$

$\1.3764=\frac{x}{2.5;cm}\x=3.44;cm$

Now that we know the apothem length, we can plug in all our values to solve for area:

$Area=5\cdot\frac{1}{2}\cdot 5;cm \cdot3.44;cm=43;cm$

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

$108^\circ$

$64^\circ$

$540^\circ$

$720^\circ$

$216^\circ$

Explanation

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: