Other Polygons

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Find the area of the shaded region:

$28 \pi\ m^2$

$26 \pi\ m^2$

$24 \pi\ m^2$

$30 \pi\ m^2$

$32 \pi\ m^2$

Explanation

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:

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

$A = 28 \pi\ m^2$

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

Explanation

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

Find the area of the shaded region:

Screen_shot_2014-03-01_at_9.02.03_pm

$9\pi - 18cm^2$

$6\pi - 18cm^2$

$9\pi - 24cm^2$

$9\pi - 9cm^2$

$6\pi - 9cm^2$

Explanation

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

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

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

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

Explanation

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

Find the area of the shaded region:

$28 \pi\ m^2$

$26 \pi\ m^2$

$24 \pi\ m^2$

$30 \pi\ m^2$

$32 \pi\ m^2$

Explanation

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:

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

$A = 28 \pi\ m^2$

Find the area of the shaded region:

Screen_shot_2014-03-01_at_9.02.03_pm

$9\pi - 18cm^2$

$6\pi - 18cm^2$

$9\pi - 24cm^2$

$9\pi - 9cm^2$

$6\pi - 9cm^2$

Explanation

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

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

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

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

Explanation

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 .

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

Explanation

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 .

Find the area of the shaded region:

Screen_shot_2014-03-01_at_9.04.49_pm

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

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

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

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

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

Explanation

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

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

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

Find the area of the shaded region:

Screen_shot_2014-03-01_at_9.04.49_pm

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

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

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

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

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

Explanation

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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