![]() The height of the pyramid is h/(1 - r), Using this in the formula for the volume of a pyramid, P = 1/3 b 1 2 h./(1 - r) ), which is a geometric series in r 3, so P = V/ (1 - r 3). The volume P of the pyramid is V(1 + r 3 + (r 3) 2 + (r 3) 3 + (r 3) 4 +. The volume of the nth figure is proportional to the cube of the scaling factor, and so will be V(r n) 3 = V(r 3) n. The scaling factor isĪgain r = b 2/b 1 and once again the scaling factor for the nth figure is r n. We have b 1 for the lower base sides and b 2įor the upper base sides. It is now relatively easy to find the volume of a square truncated pyramid using the technique we used for trapezoids. Substituting b 2 /b 1 for r, A = 1/2 b 1 h (1 + b 2/b 1) = 1/2 h ( b 1 + b 2 ), which is the standard formula for the area of a trapezoid. This idea will be more useful when we get to the three In this case, n = 1 giving the very short geometric series 1 + r. Notice that (1 - r 2)/ (1 - r) is a speical case of the formula for geometric series. R and equate the two expressions for the area of the triangle.Įquating equation 1 with equation 2 gives We could at this point substitute b 2/b 1Įxpression for the area of the triangle, but let's keep it in terms of The area of the triangle is therefore T = 1/2 b 1 H = 1/2 b 1 h /(1 - r). ![]() The scaling factor of r again applies, and so we get H = h(1 Geometric series to find the altitude H of the triangle in terms of h. Next, let's find the area of the triangle in terms of b 1 and h, using a This is just an infinite geometric series in r 2, Since the scaling factor is r n andĪrea is proportional to the square of the scaling factor, the area of The next thing we want to do is to find the area of the nth trapezoid b 3 = r 2 b 1Īnd for the nth trapezoid the scaling factor will be r n. The scaling factorĪpplies the scaling factor r to BEFC and so has a scaling factor of r 2. By taking the height of BEFC to be rh, it will be similar to ABCD. Since b 2 is the bottom base for BEFC, the scaling factor r = b 2/b 1. We want trapezoid BEFC to be similar to ABCD. The trapezoid whose area A we want to find is ABCD given the lengths ofĪBCD has b 2 for its bottom base instead of b 1. The area of the triangle allows us to solve for the area of the We will then find another geometric series to find the area We will use the geometric series to express theĪrea T of the triangle in terms of the base trapezoid. Triangle by layering copies of the trapezoid as show in the ![]() S = \dfrac = 12Ĭalculating the volume of a prism can be challenging, but with our prism volume calculator and formula, it's easy to find the volume of any prism.The strategy for finding the area of the trapezoid is to build a Here are some examples of finding the volume of a prism using the formula: Example 1įind the volume of a rectangular prism with a base of length 5 cm and width 8 cm, and a height of 10 cm.įind the volume of a triangular prism with a base of height 4 cm and base width 6 cm, and a height of 12 cm. The calculator will automatically calculate the volume of the prism.Enter the area of the base of the prism.Our prism volume calculator is designed to make it easy for you to find the volume of any prism. Where V is the volume, S is the area of the base, and h is the height of the prism. ![]() The formula for finding the volume of a prism is: Whether you are a student, a teacher, or someone who needs to work with prisms, our prism volume calculator can help you find the volume of any prism with ease. Calculating the volume of a prism is an essential skill in geometry.
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