𝑀𝐴𝐵𝐶 is a triangular pyramid whose vertex 𝑀 is at a distance of 22 centimeters from its base 𝐴𝐵𝐶. Given that the side lengths of its triangular base are three centimeters, five centimeters, and four centimeters, determine its volume.
Let’s begin with a sketch of this pyramid. And because it’s a triangular pyramid, that means that the base will be a triangle. Before we start the sketch however, let’s look at the side lengths that we’re given for the triangle: three, five, and four centimeters. We should remember that side lengths of three, four, and five in a triangle will mean that this is a right triangle because three, four, and five are one of the Pythagorean triples. When we’re drawing a diagram like this, it doesn’t have to be truly accurate or to scale. It just gives us some way in which we can represent the measurements that we’re given. Here, we’re told that vertex 𝑀 is at a distance of 22 centimeters from the base. That means that the perpendicular height from the base to the top is 22 centimeters.
For the right triangle at the base, let’s consider the lengths of three, four, and five centimeters. The longest side is the hypotenuse and that’s five centimeters. The other two sides are three centimeters and four centimeters. And so we can add these to the sketch of the pyramid. For this volume question, it doesn’t especially matter which of the line segments or sides of the triangle are three, four, or five centimeters. We’ll now consider how we can find the volume of this pyramid. To do this, we can recall that the volume of a pyramid is equal to one-third times the area of the base times the perpendicular height. However, if we try and use this formula straightaway, we do have the height of the pyramid but we don’t know the area of the base. And so, we’ll need to work it out.
Let’s return to the two-dimensional triangle which forms the base of the pyramid. We’ve already established that this is a right triangle. Taking the base as three centimeters and the height as four centimeters means that we can apply the formula that the area of a triangle is equal to one-half multiplied by the base multiplied by the height. Applying the formula to this right triangle, we have that the area of the triangle is equal to one-half times three times four. We know that three times four is 12, and half of that will give us six. And the units here are centimeters squared. So the area of the triangle is six centimeters squared.
Of course, the area of this triangle is the area of the base of the pyramid. And so we can now apply the formula to find the volume of the pyramid. This gives us that the volume of the pyramid is equal to one-third times six times 22. We can simplify before we multiply. So we have two times 22, which gives us 44. Completing the answer with the units, we have calculated the volume of this triangular pyramid as 44 cubic centimeters.