Other Polygons - Math

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Question

A regular polygon with sides has exterior angles that measure $1.5^{\circ }$ each. How many sides does the polygon have?

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Answer

The sum of the exterior angles of any polygon, one per vertex, is $360 ^{\circ }$ . As each angle measures $1.5^{\circ }$ , just divide 360 by 1.5 to get the number of angles.

$360 \div 1.5=240$

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Question

What is the interior angle measure of any regular nonagon?

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Answer

To find the angle of any regular polygon you find the number of sides , which in this example is .

You then subtract from the number of sides yielding .

Take and multiply it by degrees to yield a total number of degrees in the regular nonagon.

Then to find one individual angle we divide by the total number of angles .

$\frac{1260}{9}=140$

The answer is .

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Question

What is the measure of one exterior angle of a regular seventeen-sided polygon (nearest tenth of a degree)?

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Answer

The sum of the measures of the exterior angles of any polygon, one per vertex, is $360^{\circ}$ . In a regular polygon, all of these angles are congruent, so divide 360 by 17 to get the measure of one exterior angle:

$360 \div 17 \approx 21.2^{\circ }$

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Question

What is the measure of one exterior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?

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Answer

The sum of the measures of the exterior angles of any polygon, one per vertex, is $360^{\circ}$ . In a regular polygon, all of these angles are congruent, so divide 360 by 23 to get the measure of one exterior angle:

$360 \div 23 \approx 15.7 ^{\circ }$

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Question

What is the measure of one interior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?

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Answer

The measure of each interior angle of a regular polygon with sides is $\frac{180 (N-2)}{N}$ . We can substitute to obtain the angle measure:

$\frac{180 (N-2)}{N} = \frac{180 (23-2)}{23} = \frac{180 \cdot 21}{23}\approx 164.3^{\circ }$

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Question

What is the magnitude of the interior angle of a regular nonagon?

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Answer

The equation to calculate the magnitude of an interior angle is $\frac{180(n-2)}{n}=int\ angle$ , where is equal to the number of sides.

For our question, .

$int\ angle= \frac{180(9-2)}{9}=\frac{(180)(7)}{9}=140^o$

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Question

What is the interior angle measure of any regular heptagon?

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Answer

To find the angle of any regular polygon you find the number of sides, . In this example, .

You then subtract 2 from the number of sides yielding 5.

Take 5 and multiply it by 180 degrees to yield the total number of degrees in the regular heptagon.

Then to find one individual angle we divide 900 by the total number of angles, 7.

$\frac{900}{7}=128.6$

The answer is .

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Question

A regular polygon has interior angles which measure $162^{\circ }$ each. How many sides does the polygon have?

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Answer

The easiest way to answer this is to note that, since an interior angle and an exterior angle form a linear pair - and thus, a supplementary pair - each exterior angle would have measure $180-162 = 18^{\circ }$ . Since 360 divided by the number of sides of a regular polygon is equal to the measure of one of its exterior angles, we are seeking such that

$\frac{360}{N} = 18$

Solve for :

$\frac{360}{N}\cdot N= 18\cdot N$

$18N \div 18= 360\div 18$

The polygon has 20 sides.

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Question

What is the area of a regular heptagon with an apothem of 4 and a side length of 6?

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Answer

What is the area of a regular heptagon with an apothem of 4 and a side length of 6?

To find the area of any polygon with the side length and the apothem we must know the equation for the area of a polygon which is

First, we must calculate the perimeter using the side length.

To find the perimeter of a regular polygon we take the length of each side and multiply it by the number of sides.

In a heptagon the number of sides is 7 and in this example the side length is 6 so

The perimeter is .

Then we plug in the numbers for the apothem and perimeter into the equation yielding

We then multiply giving us the area of .

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Question

What is the area of a regular decagon with an apothem of 15 and a side length of 25?

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Answer

To find the area of any polygon with the side length and the apothem we must know the equation for the area of a polygon which is

First, we must calculate the perimeter using the side length.

To find the perimeter of a regular polygon we take the length of each side and multiply it by the number of sides.

In a decagon the number of sides is 10 and in this example the side length is 25 so

The perimeter is .

Then we plug in the numbers for the apothem and perimeter into the equation yielding $Area:of:Polygon=\frac{1}{2}(250)15$

We then multiply giving us the area of .

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Question

What is the area of a regular heptagon with an apothem of and a side length of ?

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Answer

To find the area of any polygon with the side length and the apothem we must know the equation for the area of a polygon which is

We must then calculate the perimeter using the side length.

To find the perimeter of a regular polygon we take the length of each side and multiply it by the number of sides .

In a heptagon the number of sides is and in this example the side length is so

The perimeter is 56.

Then we plug in the numbers for the apothem and perimeter into the equation yielding $Area=(\frac{1}{2})(56)(4)$

We then multiply giving us the area of .

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Question

Find the area of the shaded region:

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Answer

To find the area of the shaded region, you must subtract the area of the circle from the area of the square.

The formula for the shaded area is:

$A=A_{square}-A_{circle}$

$A = (s)^2 - \pi(r)^2$ ,

where is the side of the square and is the radius of the circle.

Plugging in our values, we get:

$A = (10m)^2 - \pi(5m)^2$

$A = 100m^2 - 25\pi m^2$

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Question

Find the area of the shaded region:

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Answer

To find the area of the shaded region, you need to subtract the area of the triangle from the area of the sector:

$A = A_{sector} - A_{triangle}$

$A = \pi r^2 (part) - \frac{1}{2} (b)(h)$

Where is the radius of the circle, is the fraction of the circle, is the base of the triangle, and is the height of the triangle

Plugging in our values, we get

$A = \pi (6cm^2) (\frac{90}{360}) - \frac{1}{2} (6cm)(6cm)$

$A = 9\pi - 18cm^2$

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Question

Find the area of the shaded region:

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Answer

To find the area of the shaded region, you need to subtract the area of the equilateral triangle from the area of the sector:

$A = A_{sector} - A_{equilateral triangle}$

$A = \pi r^2 (part) - \frac{s^2 \sqrt{3}}{4}$

Where is the radius of the circle, is the fraction of the circle, and is the side of the triangle

Plugging in our values, we get

$A = \pi (5cm^2) \frac{60}{360} - \frac{(5cm)^2\sqrt{3}}{4}$

$A=\frac{25\pi }{6}- \frac{25\sqrt{3}}{4} cm^2$

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Question

Find the area of the shaded region:

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Answer

The formula for the area of the shaded region is

$A = A_{large circle}-A_{small circle}= \pi r_{large}^2 - \pi r_{small}^2,$

where is the radius of the circle.

Plugging in our values, we get:

$A = \pi (8\ m)^2 - \pi (6\ m)^2$

$A = 28 \pi\ m^2$

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Question

Find the area of the following octagon:

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Answer

The formula for the area of a regular octagon is:

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

Plugging in our values, we get:

$A = \frac{1}{2}(96\ m)(6\sqrt{2}\ m+6\ m)$

$A = 288\sqrt{2}+288\ m^2$

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Question

Find the area of a rectangle with a base of and a width of in terms of .

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Answer

This problem simply becomes a matter of FOILing (first outer inner last)

The area of the shape is Base times Height.

So, multiplying and using FOIL, we get an area of $x^{2} - 9$

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Question

Find the area of a square whose diagonal is .

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Answer

If the diagonal of a square is , we can use the pythagorean theorem to solve for the length of the sides.

= length of side of the square

Doing so, we get $2x^{2} = 144$

$x = \sqrt{72}$

To find the area of the square, we square , resulting in .

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Question

If the area of a regular octagon is 160 and the apothem is 8, what is the side length?

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Answer

To find the side length from the area of an octagon and the apothem we must use the area of a polygon which is

First plug in our numbers for area and the apothem to get

$160=\frac{1}{2}(8)(perimeter)$

Then multiply to get

Then divide both sides by 4 to get the perimeter of the figure. $\frac{160}{4}=40$

When we have the perimeter of a regular polygon, to find the side length we must divide by the number of sides of the polygon, in this case 8. $\frac{40}{8}=5$

After dividing we find the side length is

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Question

All segments of the polygon meet at right angles (90 degrees). The length of segment $\overline{AB}$ is 10. The length of segment $\overline{BC}$ is 8. The length of segment $\overline{DE}$ is 3. The length of segment $\overline{GH}$ is 2.

Find the perimeter of the polygon.

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Answer

The perimeter of the polygon is 46. Think of this polygon as a rectangle with two of its corners "flipped" inwards. This "flipping" changes the area of the rectangle, but not its perimeter; therefore, the top and bottom sides of the original rectangle would be 12 units long $\dpi{100} \small (10+2=12)$ . The left and right sides would be 11 units long $\dpi{100} \small (8+3=11)$ . Adding all four sides, we find that the perimeter of the recangle (and therefore, of this polygon) is 46.

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