Discussion: The Prismoidal Formula

A prismatoid(a.k.a. prismoid) is a solid where all vertices lie on two parallel planes. According to the prismoidal formula, the volume can be calculated by this simple equation \[V = \frac{h}{6} \left({A}_{T} + 4{A}_{M} + {A}_{B}\right )\] where $h$ denotes the height, and ${A}_{T}$, ${A}_{M}$, ${A}_{B}$ are the top, middle and bottom cross-sectional areas respectively.

However, the prismoidal formula is not a universal formula for computing the volumes of solids. Since the prismodal formula is actually Simpson's rule, the prismatoid formula is precise if the shape is bounded by a polynomial function up to degree three. This can be proven via Lagrange error bounds Error bound of Simpson's Rule.

Check out my other notes at Proof, Disproof, and Derivation

#Geometry #Calculus #LagrangeErrorBounds #Simpson'sRule

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Comments

can you determine the parts of prismatoid?

Nezthiel Mejos - 6 years, 4 months ago

What do you mean by determine?

Steven Zheng - 6 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$