, as the length of the altitude corresponding to the hypotenuse, is the geometric mean of the lengths of the parts of the hypotenuse it forms; that is, it is the square root of the product of the two:

.

The area of , the shaded region, is half the products of its legs:

The area of  is half the product of its hypoteuse, which we can see as the base, and the length of corresponding altitude  :

comprises

of .

The area of the dirt path is the area of the outer square minus that of the inner square.

The outer square has sidelength 75 feet and therefore has area

 square feet.

The inner square has sidelength  feet and therefore has area 

 square feet.

Subtract to get the area of the dirt path:

 square feet.

The length of the garden is  feet less than that of the entire lot, or 

;

The width of the garden is  less than that of the entire lot, or 

;

The area of the garden is their product:

As an interior angle of a regular decagon,  measures

.

.

 can be found using the Law of Cosines:

First, it is necessary to determine the radius of the circle. This is the distance between  and , so we apply the distance formula:

Subsequently, the area of the circle is 

Now, we need to find the central angle of the shaded sector. This is found using the relationship

Using a calculator, we find that ; since we want a degree measure between  and , we adjust by adding , so

The area of the sector is calculated as follows:

You own a mug with a circular bottom. If the distance around the outside of the mug's base is   what is the area of the base?

Begin by solving for the radius:

Next, plug the radius back into the area formula and solve:

So our answer is:

You have a right triangle with a hypotenuse of 13 inches and a leg of 5 inches, what is the area of the triangle?

So find the area of a triangle, we need the following formula:

However, we only know one leg, so we only know b or h.

To find the other leg, we can either use Pythagorean Theorem, or recognize that this is a 5-12-13 triangle. Meaning, our final leg is 12 inches long.

To prove this:

Now, we know both legs, let's just plug in and solve for area:

You have a rectangular-shaped rug which you want to put in your living room. If the rug is 12.5 feet long and 18 inches wide, what is the area of the rug?

To begin, we need to realize two things. 

1) Our given measurements are not in equivalent units, so we need to convert one of them before doing any solving.

2) The area of a rectangle is given by: 

Now, let's convert 18 inches to feet, because it seems easier than 12.5 feet to inches:

Now, using what we know from 2) we can find our answer

The area of a triangle with two sides of lengths  and  and included angle of measure  can be calculated using the formula

.

Setting  and evaluating :

The area of a triangle, given its three sidelengths, can be calculated using Heron's formula:

,

where , , and  are the lengths of the sides, and .

Setting , , and , evaluate :

and, substituting in Heron's formula:

To the nearest whole, this is 260.