角錐體的證明

Volume of Square Pyramids

V = Bh/3 = s2h/3

General Pyramid
General Pyramid

To find the volume of a square based pyramid, we will start with a cube, which is a prism that has edges of equal length, which have been labelled with the traditional letter s.

General Cube
General Cube

Next, segments will be drawn from each vertex (corner) that will travel through the cube's center to other vertices. In doing so, six congruent (equal) pyramids are created. The volume of the cube can be found by using the formula for a prism, namely V = Bh. So, V = s2 times s = s3.

Cube with Diagonals Drawn
General Cube
with Diagonals

The volume of each of the pyramids are equal; therefore, their volumes must be 1/6 the original cube's volume. This means each pyramid has a volume, V = s3/6.

1/6 Cube or One Pyramid
1/6 Cube or
One Pyramid

By examining the cube with its diagonals, we can see that it takes two pyramid heights to be equal to the base length of the cube, or 2h = s. By substitution, we get V = s3/6 = (s2 times s)/6 = (s2 times 2h)/6. By simplifying the expression, we get V = 2(s2 times h)/6, which is the same as V = 2Bh/6 or V = Bh/3.

Example 1: If s = 3 in and h = 5 in, then the volume would be
V = Bh/3 = (3 in)2(5 in)/3 = (9 in2)(5 in)/3 = 45 in3/3 = 15 in3.

Example 2: If s = 7 m and h = 12 m, then the volume would be
V = Bh/3 = (7 m)2(12 m)/3 = (49 m2)(12 m)/3 = 588 m3/3 = 196 m3.

* Quizmaster: Volume of Pyramids

Volume of Cones

V = Bh/3 = πr2h/3

General Cone
General Cone

The formula for the volume of a cone can be determined from the volume formula for a cylinder. We must start with a cylinder and a cone that have equal heights and radii, as in the diagram below.

Cone and Cylinder of Equal Heights and Radii
Cone & Cylinder of
Equal Heights and Radii

Imagine copying the cone so that we had three congruent cones, all having the same height and radii of a cylinder. Next, we could fill the cones with water. As our last step in this demonstration, we could then dump the water from the cones into the cylinder. If such an experiment were to be performed, we would find that the water level of the cylinder would perfectly fill the cylinder.

Comparing the Volumes of Cones and Cylinders

This means it takes the volume of three cones to equal one cylinder. Looking at this in reverse, each cone is one-third the volume of a cylinder. Since a cylinder's volume formula is V = Bh, then the volume of a cone is one-third that formula, or V = Bh/3. Specifically, the cylinder's volume formula is V = πr2h and the cone's volume formula is V = πr2h/3.

This same relationship exists between pyramids and prisms. If we were to start with a pyramid and a prism with congruent bases and heights, we would find the exact same ratio of volumes. It would take three pyramids to completely fill a prism. This is why their formulas share the same relationship with that of cones and cylinders. For prisms and cylinders, their volumes are V = Bh. For pyramids and cones, their volumes are V = Bh/3.

Example 1: If r = 4 ft and h = 5 ft, then the volume would be
V = Bh/3 = πr2h/3 = (3.14)(4 ft)2(5 ft)/3 = (3.14)(16 ft2)(5 ft)/3 = 83.7 ft3.

Example 2: If r = 6 cm and h = 2 cm, then the volume would be
V = Bh/3 = πr2h/3 = (3.14)(6 cm)2(2 cm)/3 = (3.14)(36 cm2)(2 cm)/3 = 75.4 cm3.

* Quizmaster: Volume of Cones