With three-dimensional shapes, the surface area of the object is defined as the total surface area, or the sum of the areas, on every outward surface. (This is different from the lateral surface area, which excludes the base and top of 3-D shapes).
Just like the area of a two-dimensional shape, the surface area definition is measured in square units. Depending on the shape and surface of the object, there are different surface area formulas to keep in mind. We’ll go through the surface area of each shape in turn.
Calculating the Surface Area of a Rectangular Prism
The easiest way to find the total surface area of a three-dimensional shape is to find the surface area of each side and add all the surface areas. Let's take the easiest example first.
A rectangular prism has six sides or faces. Surface areas that are directly opposite to each other have the same areas, so the surface area definition of a rectangular prism is
W stands for width, l stands for length, and h stands for the height of the rectangular prism. The top surface of the rectangular prism is calculated by wl, and the two side surfaces are calculated by hl and hw respectively.
Calculating the Surface Area of a Cube
You can see from the graphic above that for a cube, all six sides are the same. So the surface area definition for a cube is A = 6a² where a stands for the length of a side.
Each face has an area of a². This is calculated by multiplying the width a by the length a of the square). There are 6 total surfaces to the cube, so we multiply by 6.
Calculating the Surface Area of a Cylinder
A cylinder has a circular base and a given height. We can calculate its surface area by:
- Splitting the surfaces into different polygons: circles for the top and bottom and a rectangle for the surface of the body (also known as its lateral surface area)
- Calculating the surface area of the rectangle and circles
- Adding the surface area of the rectangle and the circles
Here’s why we have a rectangle: Imagine if you were to cut a "slit" through the cylinder (think a toilet paper roll) and then unfold the shape. The outer surface becomes a rectangle!
First, we’ll find the rectangle’s area. We already know you can find the area of a rectangle by multiplying width times length. The length here is h, the height of the cylinder. The width here is its circumference, 2πr (r=radius). So the lateral surface area is calculated as a rectangle whose area is 2πrh.
We know the surface area of a circle is A=πr².
So the total surface area is the sum or the area of the top circle, the area of the bottom circle, and the area of the lateral surface: A = 2πr² + 2πrh
Surface Area of a Sphere
Because a sphere has a curved surface, there is no easy way to split it into different polygons as we can for cylinders and rectangular prisms. The surface area definition of a sphere is its own formula: A = 4πr²
For example, here’s how you’d calculate the surface area of a sphere whose radius is 3 feet:
Surface Area Definition for 3D Objects
For 3D objects, the definition of surface area is the total area of all its surfaces. This is easy to calculate when you know the area formula for all of its 2D polygons. Other times, like with a sphere, you are better off just memorizing the surface area formula.
Feel free to revisit this page when you can’t quite remember a surface area definition or formula — we’re here to help.