# Advanced Geometry : How to find the surface area of a cone

## Example Questions

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

What is the surface area of the following cone?

Explanation:

The formula for the surface area of a cone is:

,

where represents the radius of the cone base and represents the slant height of the cone.

Plugging in our values, we get:

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

The surface area of cone  is . If the radius of the base of the cone is , what is the height of the cone?

Explanation:

To figure out , we must use the equation for the surface area of a cone, , where  is the radius of the base of the cone and  is the length of the diagonal from the tip of the cone to any point on the base's circumference. We therefore first need to solve for  by plugging what we know into the equation:

This equation can be reduced to:

For a normal right angle cone,  represents the line from the tip of the cone running along the outside of the cone to a point on the base's circumference. This line represents the hypotenuse of the right triangle formed by the radius and height of the cone. We can therefore solve for  using the Pythagorean theorem:

so

Our  is therefore:

The height of cone  is therefore

### Example Question #1 : Surface Area

A circle of radius five is cut into two pieces,  and . The larger section is thrown away. The smaller section is curled until the two straight edges meet, and a bottom is made for the cone.

What is the area of the bottom?

Explanation:

When the smaller portion of the circle is curled in, it will make the top of a cone. The circumfrence of the circle on the bottom is  (where r is the radius of the circle on the bottom). The circumference of the bottom is also  of the circumfrence of the original larger circle, which is  (where R is the radius of the original, larger circle)

Therefore we use the circumference formula to solve for our new r:

Substituting this value into the area formula, the area of the small circle becomes:

### Example Question #1 : Solid Geometry

A cone has a bottom area of  and a height of , what is the surface area of the cone?

Explanation:

The area of the bottom of the cone yields the radius,

The height of the cone is , so the Pythagorean Theorem will give the slant height,

The area of the side of the cone is  and adding that to the  given as the area of the circle, the surface area comes to

### Example Question #1 : Solid Geometry

If the surface area of a right angle cone  is , and the distance from the tip of the cone to a point on the edge of the cone's base is , what is the cone's radius?

Explanation:

Solving this problem is going to take knowledge of Algebra, Geometry, and the equation for the surface area of a cone: , where  is the radius of the cone's base and  is the distance from the tip of the cone to a point along the edge of the cone's base. First, let's substitute what we know in this equation:

We can divide out  from every term in the equation to obtain:

We see this equation has taken the form of a quadratic expression, so to solve for  we need to find the zeroes by factoring. We therefore need to find factors of  that when added equal . In this case,  and :

This gives us solutions of  and . Since  represents the radius of the cone and the radius must be positive, we know that  is our only possible answer, and therefore the radius of the cone is .

### Example Question #2 : Surface Area

For a right circular cone , the radius is  and the height of the cone is . What is the surface area of the cone in terms of ?

Explanation:

To solve this problem, we will need to use the formula for finding the surface area of a cone, , where  is the length of the diagonal from the circle edge of the cone to the top. Since we are not given s, we must find it by using Pythagorean's Theorem:

is a prime number, so we cannot factor the radical any further. Therefore, our equation for our surface area of  becomes:

, which is our final answer.

### Example Question #1 : Cones

If the surface area of cone  is , and the distance  between the cone's tip and a point on the cone's circular base is , what is the radius  of the cone?

Explanation:

To find out the radius, we must use our knowledge of the formula for the surface area of a cone: , where  is the radius of the cone and  is the distance from the tip of the cone to any point along the circumference of the cone's base. We can plug in what we already know into the above equation:

We can divide out  from each term to obtain:

We now can recognize that the above is a quadratic expression, so to solve for  we can find the zeroes of the equation by factoring. We need two numbers which will multiply to  but will add to  (in this case  and ). Therefore, we can factor the above to the following:

.

Our two solutions are therefore  and . Since  represents the radius of the base of the cone, it must be positive, and that leaves  as our one and only answer.

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

Find the surface area of a cone with a base diameter of  and a slant height of .

Explanation:

The Surface Area of a cone is:

Given the base diameter is 6, the radius will be 3.  The given slant height is 10.

Substitute the radius and slant height into the equation to find surface area.

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

Find the surface area of a cone with a base area of  and a slant height of .

Explanation:

The surface area of a cone is:

Given the base area is , the base of the cone resembles a circle. Using the base area, it is necessary to find the radius.

Since radius of the base is 2, and slant height is 6, substitute these into the surface area equation.

### Example Question #1 : Advanced Geometry

Find the surface area of a cone with a base diameter of  and a height of .