How to find the area of a polygon - High School Math

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Question

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

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?

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 ?

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:

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:

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:

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:

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:

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 .

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 .

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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