# High School Math : How to find the area of a polygon

## Example Questions

### Example Question #1 : Plane Geometry

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

Explanation:

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  .

### Example Question #1 : How To Find The Area Of A Polygon

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

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

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

We then multiply giving us the area of  .

### Example Question #3 : Geometry

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

We then multiply giving us the area of  .

### Example Question #4 : Geometry

Find the area of the shaded region:

Explanation:

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:

,

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

Plugging in our values, we get:

### Example Question #5 : Geometry

Find the area of the shaded region:

Explanation:

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

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

### Example Question #6 : Geometry

Find the area of the shaded region:

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:

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

### Example Question #11 : Geometry

Find the area of the shaded region:

Explanation:

The formula for the area of the shaded region is

where  is the radius of the circle.

Plugging in our values, we get:

### Example Question #12 : Geometry

Find the area of the following octagon:

Explanation:

The formula for the area of a regular octagon is:

Plugging in our values, we get:

### Example Question #13 : Geometry

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

none of the other answers

Explanation:

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

### Example Question #14 : Geometry

Find the area of a square whose diagonal is .

none of these answers