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 Square Pyramids Calculator

What is your shape?

$s$

$l$

$w$

$h$

$r$

$h$

$s$

$r$

$s$

$h$

Remember: $h$ is the slanted height of the pyramid!

$s=$

$l=$

$w=$

$h=$

$r=$

$h=$

$h=$

$s=$

$r=$

$s=$

Step 1. Unwrap the square pyramid.

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 square pyramid.

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

$s$

$l$

$w$

$h$

$r$

$h$

$r$

$s$

$s$

$h$

Shape

Area

1

(Square)

$A =s_{2}=(0)_{2}=0 $

2

(Triangle)

$A =21 b×h=21 (0×0)=21 (0)=20 $

3

(Triangle)

$A =same as shape 2=20 $

4

(Triangle)

$A =same as shape 2=20 $

5

(Triangle)

$A =same as shape 2=20 $

Step 3. Add up the areas of the shapes that make up the square pyramid.

The surface area of this square pyramid is $SA=0$.

Key Steps 🗝 How to Find Surface Area of Square Pyramids

Step 1. Unwrap the cone.

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

$s$

$l$

$w$

$h$

$r$

$h$

$r$

$s$

$s$

$h$

Square

$A=s_{2}$

Triangles

$A=21 sh$

Step 3. Add up the areas of the shapes that make up the square pyramid.

$SA=s_{2} +(4×21 sh )$

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

Surface Area of Square Pyramids

The surface area of a square pyramid is equal to the sum of the areas of all the faces (sides and base) of the pyramid.

If we unwrap a square pyramid, we can see that the net is made up of $1$ square and $4$ identical triangles.

$s$

$l$

$w$

$h$

$r$

$h$

$r$

$s$

$s$

$h$

So, we need to first find the area of one of the triangles ($A=21 bh$). Then, we multiply it by $4$ and add it to the area of the square ($A=s_{2}$) to get the total surface area.$SA=s_{2} +(4×21 sh )$Notice that the side length of the square base is the same as the base of the triangle, and the slanted height of the pyramid is the height of the triangle 😮.

What about the lateral surface area?

If we wanted to find the lateral surface area of the square pyramid, we would exclude the area of the square base:

$lateral surface area of square pyramid =s_{2}+(4×21 sh )=4×21 sh $

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

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

What is your shape?

$s$

$l$

$w$

$h$

$r$

$h$

$s$

$r$

$2$

$3$

Remember: $h$ is the slanted height of the pyramid!

Step 1. Unwrap the square pyramid.

This gives us our net:

$s$

$l$

$w$

$h$

$r$

$h$

$s$

$r$

$2$

$3$

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

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 🎉.

$s$

$l$

$w$

$h$

$r$

$h$

$r$

$s$

$2$

$3$

Shape

Area

1

(Square)

$A =s_{2}=(2)_{2}=4 $

2

(Triangle)

$A =21 b×h=21 (2×3)=21 (6)=3 $

3

(Triangle)

$A =same as shape 2=3 $

4

(Triangle)

$A =same as shape 2=3 $

5

(Triangle)

$A =same as shape 2=3 $

Step 3. Add up the areas of the shapes that make up the square pyramid.

$SA =4 +3 +3 +3 +3 =4 +(4×3 )=4 +12 =16 $

The surface area of this square pyramid is $SA=16$.

Incredible work! Try another one:

What is your shape?

$s$

$l$

$w$

$h$

$r$

$h$

$s$

$r$

$3$

$4$

Remember: $h$ is the slanted height of the pyramid!

Step 1. Unwrap the square pyramid.

This gives us our net:

$s$

$l$

$w$

$h$

$r$

$h$

$s$

$r$

$3$

$4$

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

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 🎉.

$s$

$l$

$w$

$h$

$r$

$h$

$r$

$s$

$3$

$4$

Shape

Area

1

(Square)

$A =s_{2}=(3)_{2}=9 $

2

(Triangle)

$A =21 b×h=21 (3×4)=21 (12)=6 $

3

(Triangle)

$A =same as shape 2=6 $

4

(Triangle)

$A =same as shape 2=6 $

5

(Triangle)

$A =same as shape 2=6 $

Step 3. Add up the areas of the shapes that make up the square pyramid.

$SA =9 +6 +6 +6 +6 =9 +(4×6 )=9 +24 =33 $

The surface area of this square pyramid is $SA=33$.

Amazing! Practice with some additional problems here ⬇️.

Practice: Surface Area of Square Pyramids

Question 1 of 5: Find the surface area of this square pyramid:

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$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$2$

$6$

Step 1. Unwrap the square pyramid.

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$2$

$6$

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

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$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$undefined$

$2$

$6$

Since the net is made up of one square and four identical triangles, we only need to find the area of the square and one of the triangles.

Shape

Area

1

(Square)

$A$$=$$s_{2}$$=$

$_{2}$

$=$

2

(Triangle)

$A$$=$$21 s×h$$=$

$21 $$×$

$=$

3

(Triangle)

The same area as the area of shape

4

(Triangle)

The same area as the area of shape

5

(Triangle)

The same area as the area of shape

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

$SA$$=$$4 +6 +6 +6 +6 $$=$$4 +(4×6 )$$=$

$4 $$+$

$=$

Awesome job! The surface area of the square pyramid is $SA=28$.

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?