The surface area of a 3D shape is equal to the sum of the areas of all the faces (sides) of that shape. A helpful way to think about it is:

surfacearea = sum of faceareas

We can also think about surface area in terms of gifts! The surface area is like the minimum amount of wrapping paper we need to prep a gift for its surprise reveal. 🤩If we were to unwrap a gift, the amount of wrapping would be equal to the surface area of the object it was covering.When we “unwrap” a 3D shape, we get what we call a net. A surface area net shows us the different 2D shapes that make up the faces of the 3D shape.

We can then find the area of each face and then add them all together to find the total surface area.

What is lateral surface area?

When working with pyramids, cylinders, and cones, you may also be asked to find the lateral surface area.

Lateral surface area is simply the total surface area of an object minus the area of its base(s).

Surface Area of Cones Calculator

What is your shape?

$s$

$l$

$w$

$h$

$r$

$h$

$s$

$r$

$s$

$h$

$s=$

$l=$

$w=$

$h=$

$r=$

$h=$

$h=$

$s=$

$r=$

$s=$

Step 1. Unwrap the cone.

This gives us our net:

$s$

$l$

$w$

$h$

$r$

$h$

$s$

$r$

$s$

$h$

Step 2. Calculate the area of each part of the net that makes up the cone.

Sides with the same color mark are the same size 🎉.

$s$

$l$

$w$

$h$

$r$

$h$

$r$

$s$

$s$

$h$

Shape

Area

1

(Sector)

$A =πrs=π×0×0=0π $

2

(Circle)

$A =πr_{2}=π(0)_{2}=0π $

Step 3. Add up the areas of the shapes that make up the cone.

$SA =0π +0π =0π≈0 $

The surface area of this cone is $SA=0π≈0$.

Key Steps 🗝 How to Find Surface Area of Cones

Step 1. Unwrap the cone.

Step 2. Calculate the area of each part of the net that makes up the cone.

$s$

$l$

$w$

$h$

$r$

$h$

$r$

$s$

$s$

$h$

Sector

$A=πrs$

Circle

$A=πr_{2}$

Step 3. Add up the areas of the shapes that make up the cone.

$SA=πrs_{l} +πr_{2} $

Continue to explore how to find the surface area of cones in more detail ✨.

Surface Area of Cones

The surface area of a cone is equal to the sum of the area of the rolled side and area of the base of the cone.

If we unwrap a cone, we can see that the net is made up of $1$ circle and $1$ portion of a larger circle, which we call a sector.

$s$

$l$

$w$

$h$

$r$

$h$

$r$

$s$

$s$

$h$

So, we need to find the area of the circle ($A=πr_{2}$) and the area of the sector ($A=πrs$), and then add them together.$SAof cone =πrs_{l} +πr_{2} $

The area of a sector (or part of a circle) is given by the product of the slanted side and the arced side divided by 2.

We can see that the slanted side is equal to the slanted side of the cone:

And we can see that the arced side is equal to the circumference of the circle base:

The area of the sector is equal to the product of these two:

$area of sector =21 (arced side×slanted side)=21 (2πr×s)=πrs $

When we add that to the area of the circle base, we get the surface area of the cone.$SAof cone =πrs_{l} +πr_{2} $

What about the lateral surface area?

If we wanted to find the lateral surface area of the cone, we would exclude the area of the circular base:

$lateral surface area of cone =πrs +πr_{2}=πrs $

Now let's unwrap some cones to practice finding their surface area!

Let's start here - find the surface area of this cone:

What is your shape?

$s$

$l$

$w$

$h$

$r$

$h$

$3$

$2$

$s$

$h$

Step 1. Unwrap the cone.

This gives us our net:

$s$

$l$

$w$

$h$

$r$

$h$

$3$

$2$

$s$

$h$

Step 2. Calculate the area of each part of the net that makes up the cone.

Now, we'll find the area of each shape in the net. Notice that some shapes have the same area:

Sides with the same color mark are the same size 🎉.

$s$

$l$

$w$

$h$

$r$

$h$

$2$

$3$

$s$

$h$

Shape

Area

1

(Sector)

$A =πrs=π×2×3=6π $

2

(Circle)

$A =πr_{2}=π(2)_{2}=4π $

Step 3. Add up the areas of the shapes that make up the cone.

$SA =6π +4π =10π≈31.4159 $

The surface area of this cone is $SA=10π≈31.4159$.

Nice work! Try another one:

What is your shape?

$s$

$l$

$w$

$h$

$r$

$h$

$4$

$3$

$s$

$h$

Step 1. Unwrap the cone.

This gives us our net:

$s$

$l$

$w$

$h$

$r$

$h$

$4$

$3$

$s$

$h$

Step 2. Calculate the area of each part of the net that makes up the cone.

Now, we'll find the area of each shape in the net. Notice that some shapes have the same area:

Sides with the same color mark are the same size 🎉.

$s$

$l$

$w$

$h$

$r$

$h$

$3$

$4$

$s$

$h$

Shape

Area

1

(Sector)

$A =πrs=π×3×4=12π $

2

(Circle)

$A =πr_{2}=π(3)_{2}=9π $

Step 3. Add up the areas of the shapes that make up the cone.

$SA =12π +9π =21π≈65.9734 $

The surface area of this cone is $SA=21π≈65.9734$.

Amazing! Practice with some additional problems here ⬇️.

Practice: Surface Area of Cones

Question 1 of 5: Find the surface area of this cone:

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$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$4$

$2$

$undefined$

$undefined$

Step 1. Unwrap the cone.

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$4$

$2$

$undefined$

$undefined$

Step 2. Calculate the area of each shape in the net.

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$2$

$4$

$undefined$

$undefined$

Since the net is made up of one circle and one sector, we need to find the area of each.

Shape

Area

1

(Sector)

$A$$=$$πrs$$=$

$π$$×$

$=$

$π$

2

(Circle)

$A$$=$$πr_{2}$$=$

$π$$_{2}$

$=$

$π$

Step 3. Add up the area of all the shapes that make up the cone.

$SA$$=$$8π +4π $$=$

$π$

Awesome job! The surface area of the cone is $SA=12π≈37.7$.

When you feel like you've mastered this lesson, click for a celebration ⬇️!

Nice work, look at you go! Thanks for checking out this lesson ☺️🙏. Where to next?