Knowing the volume of three-dimensional objects is important because volume is one of the key measures of a solid shape. It is one way to measure size. The triangular prism shape occurs naturally in the world and is found in crystals of all types. It is also an important structural element in architecture and design.
General Solution to Calculate Volume
Draw a rectangle. Label the longer side "b" and the shorter side "a." The area of this rectangle is by definition a times b or [ab].
Construct a diagonal line from one corner of the rectangle to the opposite corner, dividing the rectangle in half. Each half is in the shape of a three-sided object called a triangle.
Select one of the triangles. The area of this triangle is by definition one half the area of the original rectangle, so the area [A] of this triangle is one half of [ab], or [ab] divided by 2. Consider this triangle the base of the prism. Since length is measured in units -- say, inches -- then area is measured in the square of those units. So, in the case of inches, [A] is measured in square inches or in^2. This triangular base is a "right" triangle because one of the interior angles is a right angle, or a 90-degree angle. There are other formulas for calculating the area of other types of triangles, but the most common formula is: area equals one half the base times the height.
Imagine the triangle of area [A] is lying flat, and imagine giving this flat triangle a thickness of 1 inch. The volume of this thick triangle is 1 inch times [A] square inches or [A] in^3. While area is measured in square units, volume is measured in cubic units, thus the 3.
Extend this 1-inch-thick triangle to 2 inches. The volume of this object is twice the previous one, or 2 inches times [A] square inches, or 2A cubic inches. Continuing on in this way allows you to see that the volume of this thick triangle is the area [A] of the base times the thickness or height [H].
An Example of Calculating a Prism's Volume
Start with a rectangle with the long side equal to 4 inches and short side equal to 3 inches. The area of the rectangle is 3 inches times 4 inches, or 12 in^2.
Draw a diagonal to divide the rectangle into two equal halves. The area of either of these triangles is one half of 12 in^2 or 6 in^2.
Take one of these triangles, call it the base and extend it vertically to 12 inches. The volume of this triangular prism equals the area of the base of the prism times its height, or 6 in^2 times 12 inches, which equals 72 in^3.
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