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Finding the Volume of a (3D) Shape or Object

Finding the volume of any shape is the equivalent of finding out "how much liquid it can hold." This requires a bit more calculation than finding the area of a shape, because we need to be aware of THREE values (since the volume implies a 3D shape.) These values are the width, length, and height. (In computer science and higher math, these 3 values are also referred to as "x, y, and z.") Let's get started.

Find the volume of a cube (3d square) or box (3d rectangle)
Since finding the area of a 2-Dimensional Square or Rectangle involves using two values, it would seem that finding the 3D Area (also known a the volume) would require 3 values. And luckily, it is just that easy. Squares and rectangles are flat - they have no height. Cubes and Boxes (3D) have a height, so we add that into our calculations.

Formula: Volume of a Cube or Box = Length * Width * Height

Given the above cube, the height is 4, width is 3, and length is 5. We multiply all these, and discover the volume is 60!


Finding the volume of a Sphere (3D Circle)
This is one of the more weird formulas to remember. But it's also one of the more useful in various areas. Without further ado, the formula you've all been waiting for:

Formula: Volume of a Sphere = 4/3 pi * r³

We use radius cubed since volume is a 3D (cubic) measurement.

Here we have a beautiful blue sphere, with a radius of 4 inches. Thus, our formula to determine its volume is: 4/3 * 3.14 * 4³. That's 4/3 * 3.14 * 64. Pop that into a calculator (or do it longhand), and the volume is 267.2768 cubic inches.

Finding the Volume of a Cone
A cone looks like a 3D triangle of sorts. But if you strategically line three of them up in a line, the make a cube! Using this knowledge, you can find the volume of a cone by using the almost same formula of the cube, but since it takes 3 cones to make 1 cube, we can multiple by 1/3 (or divide by 3; it's the same thing afterall!)

Formula: Volume of a Cone = 1/3 * base * height.

Using the above cone, we can find the volume given the base of 6 and height of 5. The volume is 1/3 * 6 * 5, which equals 10.

Finding the Volume of a Cylinder
A cylinder is basically a cross-section of a circle, extended up and down. Because of this relationship to a circle, we can use some of what we already know about circles to determine the volume.

Formula: Volume of a Cylinder = pi * height * radius²

(Note: This the same as the area of a circle, except we've introduced a 3rd dimension, height!)

With this cylinder, with a height of 5 inched and radius of 3 inches, we determine the volume by mixing in pi. The equation is 3.14 * 5 * 3². That's 3.14 * 5 * 9, giving us the total volume of 141.3 cubic inches.


That's all there is too it! It may seems confusing since there are so many shapes, but if you understand the relationships (Cylinders to Circles, squares to Cubes, for example), it makes remembering the formula much easier!

Now on to finding the Surface Area of shapes! Or select another Math Lesson from our Lesson Center.



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