# High School Math : Geometry

## Example Questions

### Example Question #131 : High School Math

What is the perimeter of a regular hendecagon with a side length of 32?

Explanation:

To find the perimeter of a regular hendecagon you must first know the number of sides in a hendecagon is 11.

When you know the number of sides of a regular polygon to find the perimeter you must multiply the side length by the number of sides.

In this case it is .

The answer for the perimeter is

### Example Question #1 : Geometry

All segments of the polygon meet at right angles (90 degrees). The length of segment is 10. The length of segment is 8. The length of segment is 3. The length of segment is 2.

Find the perimeter of the polygon.

Explanation:

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 . The left and right sides would be 11 units long . Adding all four sides, we find that the perimeter of the recangle (and therefore, of this polygon) is 46.

### Example Question #2 : Geometry

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

Explanation:

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 nonagon the number of sides is , and in this example the side length is .

The perimeter is .

### Example Question #3 : Geometry

Find the perimeter of the following octagon:

Explanation:

The formula for the perimeter of an octagon is .

Plugging in our values, we get:

### Example Question #4 : 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 #5 : Geometry

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 #6 : 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 #7 : 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 #8 : 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 #9 : Geometry

Find the area of the shaded region: