Ever looked at an ice cream cone and thought, “Hey, I wonder how much surface area that has?” Well, guess what—there’s actual math behind that! In geometry, the surface area of a cone is all about how much space is covered by its surface, including both the flat circular base and the curved part that wraps around.

Here’s the deal:

• The Total Surface Area (TSA) of a cone = the area of the base plus the area of the curved surface.
• The formula? It’s given by:
  $\text{Total Surface Area} = \pi r (r + l)$
  where:
  • $r$ = radius of the base
  • $l$ = slant height of the cone (that diagonal length up the side)

Now, if you're only interested in the curved part—say, you're calculating the amount of paper needed to wrap around a cone-shaped party hat—you’ll use the Curved Surface Area (CSA) formula:
$\text{Curved Surface Area} = \pi r l$

A cone itself is a super cool three-dimensional shape. Think of it like this: Take a right triangle, spin it around one of its sides, and bam—you've got a cone! You see cones everywhere: ice cream cones, funnels, traffic cones... the list goes on.

In this article, we’re going to break down everything you need to ace this concept on tests like the SAT, ACT, PSAT/NMSQT, GED, GRE, GMAT, AP Exams, PERT, Accuplacer, and even the MCAT. We’ll cover:

• What exactly is the surface area of a cone
• The key formulas (yes, the ones you need to memorize!)
• Easy-to-follow derivations
• A bunch of solved examples that'll boost your confidence

Let’s dive in and make sure you're ready to crush any surface area question that comes your way on test day!

Surface Area of a Cone

Cones are pyramid-like structures. It has a circular base that tapers at a top point known as the vertex and is a three-dimensional shape. The area occupied by a cone’s perimeter or surface is referred to as its surface area. The curved surface area and the overall surface area of a cone are the two different types of surface areas.

Square units like square meters, square centimeters, or equivalents are used to measure surface areas. A cone’s curved surface area only encompasses the curved portion. The sum of a cone’s curved surface area and base circle area is the cone’s total surface area.

Surface Area of a cone Formula

The curved surface area and the total surface area of a cone are the two different types of surface area.

Curved Surface Area of a Cone

The area of the cone’s curved portion is referred to as the cone’s curved surface.

Curved Surface Area of a Cone (S) = \(\pi rl\)

Where,

“r” = Radius of the base of a cone

“l” = Slant height of the cone.

Total Surface Area of a Cone

The total surface area of a cone is the total area that the cone occupies in two dimensions, including the area of the circular base and the area of the curving surface.

Total Surface Area of a Cone = \( \pi r\left ( r +l \right )\)

Where,

r = The radius of the base of a cone and

l = The slant height of the cone.

Derivation of the Surface area of cone

Take a paper cone and cut it at the height of the slant to see the shape the cone’s surface makes.

Mark A and B as the two endpoints, and O as the point where the two lines converge. Now, when we open this, it will resemble a circle sector.

Therefore, we have to determine the sector area in order to determine the cone curved surface area.

Area of the sector in term of arc length =\( \frac{arc length\times radius}{2}

Curved Surface Area of a Cone (S) = \(\pi rl\)

Total surface area of cone (T) = Area of the base + curved surface area

Area of the base = \(\pi r^{2}\)

\(\Rightarrow T = \pi r^{2} +\pi rl = \pi r\left ( r +l \right )\)

Surface area of Right Circular Cone

An axis that is perpendicular to the base plane defines a right circular cone. By rotating a right triangle around one of its legs, we may create a right cone. The region of space occupied by the right circular cone’s surface is known as the surface area of the cone.

Right circular cone(Curved surface area) C.S.A = \( \pi r s\)

Right circular cone(Total surface area ) T.S.A =\( \pi r^{2} + \pi r s\)

Where,

r = radius

s = slant height

Surface area of oblique Cone

A cone with a surface made up of lines connecting a fixed point to a circle points, with the fixed point’s centre sitting on a line that isn’t perpendicular to the circle.

The following formula can be used to calculate the surface area of an oblique cone:

Lateral area (LA) = \( \pi rl\)

Total area (TA) = \(\pi rl + \pi r^{2}\)

Surface area of Partial Cone

The axis of a partial cone runs through the centers at right angles to both of the circular faces of a partial cone, which has two circular faces.

The area of the larger sector (with radius) less the area of the smaller sector gives the lateral surface area of a partial cone.

Lateral area

F =\( \pi \left ( r_{1} + r_{2}\right )\sqrt{\left ( \left ( r_{1} -r_{2} \right )^{2} \right ) + h^{2}}\)

Total area (TA) =

S = F + \( \pi \left ( \left ( r_{1} \right )^{2}+\left ( r_{2} \right )^{2}\right )\)

Surface area of cone frustum

The portion of a cone that remains after being split in half by a plane is called the frustum. Cone upper portion maintains its shape, but its lower portion forms a frustum.

A cone top is cut off, keeping the cut parallel to the base, to form a cone-shaped frustum.

Let s be the slant height and \(R_{1}\) and \(R_{2}\) be the base and top radii for a right circular cone. Then

s = \(\sqrt{\left (\left ( R_{1} – R_{2} \right )^{2}\right ) + h^{2}}\)

Without considering the top and bottom circles, the surface area is

A = \( \pi \left ( R_{1} +R_{2} \right )s\)

A = \( \pi \left ( R_{1} +R_{2} \right )\sqrt{\left ( R_{1}-R_{2} \right )^{2}+h^{2}}\)

Surface area of cone by integration

https://blogmedia.testbook.com/blog/wp-content/uploads/2022/12/surface-area-of-cone_8-ad66a04d.png

The cone is shown in cross-section in the image above.

According to the apex, the area of the strip of width dh that corresponds to h is \( 2\pi r \frac{dh}{\cos x}\)

Now, \( r = h\tan x\)

\(dA = 2\pi h\frac{\tan x}{\cos x} dh\)

So, the total area

\( =\int_{H}^{0} 2\pi h\frac{\tan x}{\cos x} dh = \pi H^{2}\frac{\tan x}{\cos x} = \pi \left ( H\tan x \right )\left ( \frac{H}{\cos x} \right ) = \pi RS\)

Solved Examples

Problem: 1If a cone’s base diameter is 34 meters and its slant height is 21 meters, calculate its total surface area.

Solution:

It’s given,

Diameter of cone = 26 meters

Radius of cone = \frac{26}{2} = 13 meters

The slant height of the cone = 21 meters

As we already know that

Total Surface Area of a Cone = \( \pi r\left ( r +l \right )\)

= \(\frac{22}{7}\times 13\left ( 21 + 13 \right )\)

=\( \frac{22}{7}\times 13\times 34\)

Total Surface Area of a = \(1389.14 m^{2}\)

Problem: 2If a cone has a radius of 14 units and a curved surface area of 1100 square units, what is its height?

Solution:

Radius of cone r = 14 units

Curved surface area of the cone = 1100 square units

Let “l” represent the cone’s slant height and “h” represent the cone’s height.

Curved Surface Area of a Cone (S) = \(\pi rl\)

\(\Rightarrow 1100 = \frac{22}{7}\times 14\times l\)

\(\Rightarrow 44 \times l = 1100\)

\(\Rightarrow l = \frac{1100}{44} = 25\)

As we know that

Slant height (l) = \( \left ( l \right ) = \sqrt{\left ( h^{2} + r^{2} \right )}\)

\( \Rightarrow h = \sqrt{\left ( l^{2}-r^{2} \right )} \)

\(= \sqrt{\left ( 25^{2}-14^{2} \right )} \)

\(= \sqrt{429} \)

= 20.71 units

As a result the height of the cone is 20.71 units.

Conclusion

Understanding the surface area of a cone is essential for solving a variety of geometry problems, especially in standardized tests like the SAT, ACT, GRE, and more. By mastering both the curved and total surface area formulas, along with their derivations, you’ll gain a solid foundation in geometry. This knowledge not only helps in exams but also has practical applications in real-life scenarios involving cone-shaped objects. Keep practicing to strengthen your skills!