Pentagons - High School Math

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Question

What is the interior angle measure of any regular pentagon?

Answer

To find the angle of any regular polygon you find the number of sides . For a pentagon, .

You then subtract 2 from the number of sides yielding 3.

Take 3 and multiply it by 180 degrees to yield the total number of degrees in the regular pentagon.

Then to find one individual angle we divide 540 by the total number of angles 5, $\frac{540}{5}=108$

The answer is .

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Question

What is the measure of each interior angle of a regular pentagon?

Answer

The following equation can be used to determine the measure of an interior angle of a regular polygon, where $\small n$ equals the number of sides.

$\angle=\frac{180(n-2)}{n}$

In a pentagon, $\small n=5$ .

$\angle=\frac{180(5-2)}{5}$

Now we can solve for the angle.

$\angle=\frac{180(3)}{5}=\frac{540}{5}=108^o$

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Question

Find the interior angle of the following regular pentagon:

Answer

The formula for the sum of the interior angles of a polygon is

$X=(s-2)(180^{\circ})$ ,

where is the number of sides in the polygon

Plugging in our values, we get:

$X=(5-2)(180^{\circ})$

$X=540^{\circ}$

Dividing the sum of the interior angles by the number of angles in the polygon, we get the value for each interior angle:

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

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Question

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

Answer

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

$\angle=\frac{180(n-2)}{n}$

$\angle=\frac{180(5-2)}{5}=\frac{180(3)}{5}=108$

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Question

What is the area of a regular pentagon with side length ?

Answer

The area of a regular polygon can be solved using the following equation: $area=\ \frac{1}{2}(perimeter)(apothem)$ .

To find the perimeterof the pentagon, we multiply the side length by five.

$perimeter=5(side\ length)=5*7=35$

The apothem of a regular polygon ( = number of sides; = side length) is given by the equation below.

$apothem= \frac{s}{2tan(\frac{180}{n})}$

Pluggin in our numbers, we can solve for the apothem.

$apothem==\frac{7}{2tan(\frac{180}{5})}=4.8$

Finally, we can solve for the area.

$area= \frac{1}{2}(35)(4.8)=84$

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Question

Find the area of the following pentagon:

Answer

The formula for the area of of a pentagon is

$A=\frac{1}{2}5(side)(apothem)$ .

Plugging in our values, we get:

$A=\frac{1}{2}5(4\ m)(2.75\ m)$

$A=27.5\ m^2$

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Question

Find the area of the following pentagon:

Answer

The formula for the area of of a pentagon is:

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

Plugging in our values, we get:

$A = \frac{1}{2}5(6\ m)(3\ m)$

$A=45\ m$

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Question

Find the length of the diagonal of the following pentagon:

Answer

Use the Pythagorean Theorem to find the length of the diagonal:

$(\frac{1}{2}side)^2+(apothem)^2=(diagonal)^2$

$(2\ m)^2+(2.75\ m)^2 = C^2$

$C\approx 3.4\ m$

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Question

Find the length of the diagonal of the following pentagon:

Answer

Use the Pythagorean Theorem to find the length of the diagonal:

$(\frac{1}{2}side)^2+(apothem)^2=(diagonal)^2$

$(3\ m)^2+(3\ m)^2=C^2$

$C=3\sqrt{2}\ m$

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Question

What is the side length of a regular pentagon with a perimeter of ?

Answer

To find the side length of a regular pentagon with a perimeter of you must use the equation for the perimeter of a pentagon.

The equation is

Plug in the numbers for perimeter and number of sides to get

Divide each side of the equation by the number of sides to get the answer for the side length. $\frac{80}{5}=\frac{(side: length)*(5)}{5}$

The answer is $\frac{80}{5}=16$ .

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Question

Find the length of the side of the following pentagon.

The perimeter of the pentagon is $20\ m$ .

Answer

The formula for the perimeter of a regular pentagon is

,

where represents the length of the side.

Plugging in our values, we get:

$20\ m = 5(s)$

$s=4\ m$

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Question

Find the length of the side of the following pentagon.

The perimeter of the pentagon is $30\ m$ .

Answer

The formula for the perimeter of a regular pentagon is

,

where represents the length of the side.

Plugging in our values, we get:

$30\ m=5(s)$

$s=6\ m$

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Question

What is the perimeter of the pentagon?

Answer

The perimeter of a polygon is found by calculating the sum of all of the side lengths. In this instance, the polygon is regular, so the perimeter can be found by mulitplying the length of one side by the total number of sides:

$P=l\times n=6\times 5=30$

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