While "prism" makes most people think of a lens that separates white light into the colors of the rainbow, in geometry a "prism" is a three-dimensional figure (polyhedron) that has two faces that are the same size and shape (congruent) and has parallelograms (usually rectangles) for its other faces. Prisms are named for the shape of their congruent faces; the most common types are rectangular and triangular prisms. (The light-splitting prism depicted on Pink Floyd's cover of "Dark Side of the Moon" is a triangular prism, while a cube is a form of rectangular prism, as the squares making its faces are special cases of rectangles.) Finding the height of a prism (the distance between its congruent faces) is a matter of knowing and re-working the formula for finding its area; the steps below show how to do this for rectangular and triangular prisms.
Finding the Height of a Rectangular Prism
1Know the formula for the volume of a rectangular prism. The formula is V = Ah, where A represents the area of the base and h is the height. As we're defining the base of the prism to be one of the congruent rectangular faces, its area can be defined as the product of its length times its width, so that the volume formula can be rewritten as V = lwh, where l represents the length and w represents the width.
2Rewrite the formula to find the height. By dividing both sides of the formula by lw for the length and width, we have V/lw = lwh/lw, which becomes h = V/lw, or the height equals the volume divided by the length and width.
3Obtain the values for the volume (V), length (l), and width (w). For this example, we'll assume a volume (V) of 64, a length (l) of 8, and a width (w) of 2.
- Properly speaking, the numbers above are measurements and should be accompanied by units of measure. When a real-world measurement isn't specified, the indefinite term "units" is used. The length would thus be 8 units, and the width would be 2 units. The volume, being a 3-dimensional measurement, would be 64 cubic units.
4Find the product of the length and width. For a length of 8 and a width of 2, this is 16. This value is the area of one of the bases.
- Technically, the area is 16 square units.
5Divide the volume by the product of the length and width (area) of the base. This is 64 divided by 16, or 4 units. (Height is also a distance measure.)
- The volume could also have been divided by first the length (64 divided by 8, making 8) and then by the width (8 divided by 2), again making 4 units.
Finding the Height of a Triangular Prism
1Know the formula for the volume of a triangular prism. Again, the basic formula is V = Ah, or volume equals the area of the base times the height. However, the formula for the area of a triangle is A = 1/2 ab, where a is the distance from 1 of the triangle's vertices to its opposite side, and b is the length of that side (its base). By substituting this into the equation for A, the formula for the volume of a triangular prism is V = 1/2 abh.
- You may be more familiar with the triangle area formula as A = 1/2 bh, where b is the base of the triangle, and h refers to its height. Because we're using the terms "base" and "height" in this article for a 3-dimensional image, and using "h" to represent the height of the prism, we're using "a" in place of "h" to represent the distance from a vertex to the side opposite it (the height of the triangle).
2Rewrite the formula to find the height. In this case, we divide both sides of the equation by 1/2 ab, making V/(1/2 ab) = (1/2 abh)/(1/2 ab). This makes the formula h = V/(1/2 ab), but because dividing by 1/2 is the same as multiplying by 2, we can simplify the formula to h = 2V/ab.
3Obtain the values for the volume (V), distance from the triangle vertex to the opposite side (a), and length of the opposite side (b). For this example, we'll use a volume (V) of 840 cubic units, a distance (a) of 12 units, and a length (b) of 7 units.
4Find the product of a and b. Multiplying 12 by 7 gives a product of 84.
5Divide this product into twice the volume. Multiplying 840 by 2 equals 1680; dividing by 84 gives a value of 20 units for h.
- The height of a triangular prism should not be confused with its slant height, which is the distance from 1 of the base triangle's vertices at its base to the vertex opposite the baseline. (Using the prism on the cover of "Dark Side of the Moon" as an example again, it would be the distance from the vertex at the top of the pictured prism to either of the vertices at the bottom.) You can find the slant height by knowing the length of the base side, dividing it by 2, and the distance from the opposite vertex to the base side, then using the Pythagorean theorem, a^2 + b^2 = c^2, where c would be the value of the slant height.
Things You'll Need
- Pen/pencil and paper or calculator (optional)
Sources and Citations
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