# Basic Geometry : How to find the length of a radius

## Example Questions

### Example Question #171 : Circles

Find the length of the radius of a circle inscribed in a square that has a diagonal of .

Explanation:

Notice that the diagonal of the square is also the hypotenuse of a right isosceles triangle whose legs are also the sides of the square. You should also notice that the diameter of the circle has the same length as that of a side of the square.

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

Now, substitute in the value of the diagonal to find the length of a side of the square.

Simplify.

Now keep in mind the following relationship between the diameter and the side of the square:

Recall the relationship between the diameter and the radius.

Substitute in the value of the radius by plugging in the value of the diameter.

Solve.

### Example Question #21 : How To Find The Length Of A Radius

Find the length of the radius of a circle inscribed in a square that has a diagonal of .

Explanation:

Notice that the diagonal of the square is also the hypotenuse of a right isosceles triangle whose legs are also the sides of the square. You should also notice that the diameter of the circle has the same length as that of a side of the square.

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

Now, substitute in the value of the diagonal to find the length of a side of the square.

Simplify.

Now keep in mind the following relationship between the diameter and the side of the square:

Recall the relationship between the diameter and the radius.

Substitute in the value of the radius by plugging in the value of the diameter.

Solve.

### Example Question #182 : Basic Geometry

Find the length of the radius of a circle inscribed in a square that has a diagonal of .

Explanation:

Notice that the diagonal of the square is also the hypotenuse of a right isosceles triangle whose legs are also the sides of the square. You should also notice that the diameter of the circle has the same length as that of a side of the square.

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

Now, substitute in the value of the diagonal to find the length of a side of the square.

Simplify.

Now keep in mind the following relationship between the diameter and the side of the square:

Recall the relationship between the diameter and the radius.

Substitute in the value of the radius by plugging in the value of the diameter.

Solve.

### Example Question #21 : How To Find The Length Of A Radius

Find the length of the radius of a circle inscribed in a square that has a diagonal of .

Explanation:

Notice that the diagonal of the square is also the hypotenuse of a right isosceles triangle whose legs are also the sides of the square. You should also notice that the diameter of the circle has the same length as that of a side of the square.

In order to find the radius of the circle, we need to first use the Pythagorean theorem to find the length of the side of the square.

Now, substitute in the value of the diagonal to find the length of a side of the square.

Simplify.

Now keep in mind the following relationship between the diameter and the side of the square:

Recall the relationship between the diameter and the radius.

Substitute in the value of the radius by plugging in the value of the diameter.

Solve.

### Example Question #184 : Basic Geometry

Find the radius of a circle given the circumference is .

Explanation:

To solve, simply use the formula for the circumference of a circle and solve for r. Thus,

### Example Question #185 : Basic Geometry

Find the length of the radius of a circle given the diameter is 3.

Explanation:

To solve, simply use the formula for the diameter and solve for r. Thus,

Therefore, when solved for r,

Plug in d and:

### Example Question #181 : Basic Geometry

A circle has a circumference of  inches. What is the radius of the circle?

Explanation:

The circumference of a circle is given by , where  is the circumference and  is the radius.

Plug in the given circumference for and solve for :

### Example Question #182 : Basic Geometry

Oscar's bike will travel for the wheels on his bike to make one complete rotation. What is the radius of the wheels on Oscar's bike? Round to the nearest inch?

Explanation:

The distance that the bicycle travels for one full rotation of the wheels represents the circumference of the wheel. Therefore, if is travelled for the wheel to rotate once completely, then the circumference for the bike wheel is . With this information, we can use our circumference formula to determine the radius of the wheel:

Since radius is half the diameter, we can conclude that the radius of the wheel can be rounded to .

### Example Question #183 : Basic Geometry

What is the radius of a circle whose area is ?

Explanation:

The formula to calculate area is,

.

We are told that the area is  so we set the

diving each side by  gives us

take the square root of both sides and we see that

### Example Question #184 : Basic Geometry

Michael's Clock has an area of . Find the radius.

Explanation:

The equation to find any circle is

The total area is 169 inches^2

Divide each side by  to get rid of it on both sides. Your new equation should look like this.

Now, take the square root of each side. The square root of 169 is 13, so...