Our world is filled with three-dimensional objects. When it comes to knowing the volume and surface area of these objects, there are two definitions that you have to know. Surface area is the area of all outer facing surfaces on an object. The total surface area is calculated by adding all the areas on the surface: the areas of the base, top, and lateral surfaces (sides) of the object. This is done using different area formulas and measured in square units.
Volume is the amount of space that a three dimensional object takes up. There are also different formulas for different three-dimensional shapes. The total volume of an object is measured in cubic units.
Common Formulas for Volume and Surface Area
We have a cheat sheet for you — volume and surface area formulas for common shapes. Use it wisely!
Surface Area Formulas
The surface area of a cube is 6s², where s is the length of a side.
The surface area of a rectangular prism is 2(wl + hl + hw), where w is the width, h is the height, and l is the length.
The surface area of a sphere is 4πr², where r is the radius of the sphere.
The surface area of a cylinder is 2πrh + 2πr², where r is the radius of the cylinder and height is the height.
The surface area of a cone is πrs + πr², where r is the radius of the cone and s is the slant.
The volume of a cube is s³, where s is the length of a side.
The volume of a rectangular prism is wlh, where w is the width, h is the height, and l is the length.
The volume of a sphere is (4πr³) / 3, where r is the radius of the sphere.
The volume of a cylinder is πr²h, where r is the radius of the cylinder and height is the height.
The volume of a cone is (πr²h) / 3, where r is the radius of the cone and s is the slant.
Example of Calculating Volume and Surface Area
To find surface areas of objects with a curved surface, such as a sphere, there is no choice but to memorize the volume and surface area formulas. But for other objects, we can often break them down into other recognizable polygons and shapes whose volume or surface area we can easily find. For example, with a pyramid, you simply calculate the surface area of the base and add that to the surface area of each triangular side.
Let's find both the surface area and the volume of a square pyramid with a base length of 6 inches and a slant height of 5 inches.
Surface Area of a Square Pyramid
- To find the total surface area, we‘ll first examine the area of the base, which is just a square. We know that the area of a square is b², where b is the length of the side. In this case, the surface area is 6², or 36 square inches.
- Next, we look at the four sides, which are just triangles. We know the area of a triangle is just (bh) / 2, where b is the base and h is the height. In this case, the base of the triangle is the same as the base of the pyramid, b. The height of the triangle is equal to the slant height of the pyramid, s. So the area of just one of these triangular sides is equal to:
- When we add four of those triangles with the area of the square base, we have a total surface area of b² + 4(bs) / 2, or b² + 2bs for the square pyramid. Plugging in the numbers gives us:
Volume of a Square Pyramid
The volume of a pyramid follows this volume formula:
We can plug in our values for the height and base to give us:
Recognizing Volume and Surface Area
Again, surface area measures the area of the total outside surfaces of an object, while volume measures the internal space that the object takes up.
You’ll find many real-life cases when calculating the surface area or volume a shape would be useful, such as the amount of water it takes to fill up a pool (rectangular prism) or the amount of wrapping paper it takes to wrap a candle (cylinder) or basketball (sphere). But there are always formulas for the most common shapes. While it’s helpful to walk through the formulas before you need to use them, don't feel like you need to memorize them all! Just bookmark this list for when you need it.