--- title: "Prismoidal formula for volume" subject: "Surveying I" chapter: 9 key: "semester_3/surveying_i:prismoidal-formula-volume" source: https://www.bctengineer.com/q/semester_3/surveying_i/prismoidal-formula-volume site: BCT Engineering --- # Prismoidal formula for volume Surveying I · Chapter 9: Computation of Area and Volume - Asked in: 2068 Chaitra, 8 marks - Video: [Engineering Volumes by prismoidal formula a knec paper for building & civil engineering learners](https://www.youtube.com/watch?v=hgpu6M7V_L4) — Iwords Shots Empire - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/surveying_i/prismoidal-formula-volume - Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-9 ## Question Work out the prismoidal formula to calculate volume. ## Short answer Prismoidal formula for two parallel end areas A1, A2, length L, and mid-section area Am is V = L/6 (A1 + 4Am + A2). For multiple equal intervals, apply Simpson's rule. Prismoidal correction Cp = (L/6)(A1 + A2 - 2Am) equals trapezoidal minus prismoidal volume. ## Answer A **prismoid** is a solid having two parallel end faces (the two end sections, areas $A_1$ and $A_2$) joined by plane or ruled surfaces. The earthwork between two cross-sections is generally treated as a prismoid. The **prismoidal formula** gives its volume accurately. #### Derivation Let $L$ be the distance between the end sections, and let $x$ be the distance measured from the first end. Since each linear dimension of the section varies linearly with $x$, the cross-sectional area at distance $x$ is a second-degree (quadratic) function of $x$: $$ A(x) = a + b x + c x^2 $$ The areas at the ends and at the middle are $$ A_1 = A(0) = a,\quad A_m = A\left(\tfrac{L}{2}\right) = a + \frac{bL}{2} + \frac{cL^2}{4},\quad A_2 = A(L) = a + bL + cL^2 $$ The volume is $$ V = \int_0^L A(x)\,dx = aL + \frac{bL^2}{2} + \frac{cL^3}{3} $$ Now $A_1 + 4A_m + A_2 = 6a + 3bL + 2cL^2$, so $$ \frac{L}{6}\left(A_1 + 4A_m + A_2\right) = aL + \frac{bL^2}{2} + \frac{cL^3}{3} = V $$ $$ \boxed{V = \frac{L}{6}\left(A_1 + 4A_m + A_2\right)} $$ where $A_m$ is the area of the **middle section** (not the mean of $A_1$ and $A_2$; its dimensions are the means of the end dimensions). #### Prismoidal rule for a number of sections If there is an **odd number of sections** at equal spacing $d$ along the line (that is, an even number of intervals), apply the formula to each pair of intervals (length $2d$, with the section between them as the middle section): $$ V = \frac{d}{3}\left[A_1 + A_n + 4(A_2 + A_4 + \dots) + 2(A_3 + A_5 + \dots)\right] $$ This is the same as Simpson's one-third rule. If the number of sections is even, the last interval is treated by the trapezoidal formula. #### Prismoidal correction The trapezoidal (average end area) formula gives $V_t = \frac{L}{2}(A_1 + A_2)$. The difference between the trapezoidal and prismoidal volumes is the **prismoidal correction**: $$ C_p = V_t - V_p = \frac{L}{3}\left(A_1 + A_2 - 2A_m\right) $$ For a level section with side slope $s:1$ and end depths $h_1$ and $h_2$, this becomes $$ C_p = \frac{s\,L}{6}\,(h_1 - h_2)^2 $$ For level sections it is always positive, so the prismoidal volume = trapezoidal volume - $C_p$. The correction is large only where the depths $h_1$ and $h_2$ differ greatly. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is a prismoid? A. A solid with one end only B. A cone C. A solid with equal areas always D. A solid with two parallel end faces joined by plane or ruled surfaces Answer: D. Earthwork between two cross-sections is treated as a prismoid. (Key point: Prismoid definition) 2. What assumption allows the prismoidal formula to be exact? A. Area varies as a quadratic function of distance B. Area is constant C. Area varies linearly D. Area varies cubically Answer: A. A(x) = a + bx + cx^2; integration gives V = L/6 (A1 + 4Am + A2). (Key point: Quadratic area) 3. What is Am in the formula? A. Mean of A1 and A2 B. Area of the middle section, built from mean dimensions C. Area at the end D. Zero Answer: B. Am uses the mean of the end dimensions, not the mean of the end areas. (Key point: Middle section) 4. The prismoidal rule for several sections requires what? A. An even number of sections B. Unequal spacing C. An odd number of sections (even intervals) D. Only two sections Answer: C. It is Simpson's one-third rule and needs an even number of intervals. 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