# Trapezoidal formula for volume in surveying

JavaScript seems to be disabled in your browser. You must have JavaScript enabled in your browser to utilize the functionality of this website. Save For Later Print. Updated: October 29, The importance of getting an accurate estimation of your pond surface area cannot be overestimated.

The majority of pond owners visually estimate their pond area, which usually results in an overestimate of the true pond surface area. Pond area and water volume should be calculated based on some simple measurements. The effort necessary to estimate pond surface area is directly related to your pond's shape and uniformity.

The simplest method--using basic equations for common shapes--can be applied if your pond closely resembles a circle, square, rectangle, or trapezoid in shape. Square the shoreline distance and divide byto get the pond area in acres.

Multiply the length times the width to get the square feet of surface area. Many ponds may be roughly rectangular in shape, but one side may be significantly shorter than the other. The area of this shape is best estimated using a formula for a trapezoid by taking the average length of the two unequal sides and multiplying by the width of the pond. Many ponds have an irregular shape where the surface area cannot be adequately estimated using the formulas for common geometric shapes.

Three methods can be used in this case depending on the degree of accuracy you desire. Keep in mind that the accuracy of your pond surface area estimate may be very important, especially for the safe use of aquatic herbicides. The three methods are described in order from least to most accurate.

You should strive to use the most accurate method that you can reasonably accomplish. A pond surface area could be estimated by walking the perimeter of the pond and stopping at various waypoint locations along the pond shoreline. If waypoints are stored at each location where the pond shape changes, the resulting area will be extremely accurate, probably within 1 percent of the actual pond area.

Even if you do not own a GPS system, friends or family members that enjoy outdoor recreation may own a unit that could be used to estimate your pond surface area. The volume of water in ponds is often expressed in units called "acre-feet.

To calculate the acre-feet of water in a pond, you'll need the surface area in acres as calculated above and an average depth of water in the pond. For a typical bowl-shaped pond the average depth can be estimated as 0. So, a pond with a maximum depth of 12 feet would have an average depth of about 4. A more accurate method for calculating average depth is to make many measurements and calculate an average. This is most often done by measuring the pond depth along two transects--one along the width and one along the length.

Make sure to pick transects that represent the shallow and deep portions of the pond. Depths can be measured easily from a canoe or boat using a weight and a string marked in feet. The more depth measures that you make, the more accurate your final average will be. In the example shown on the next page, pond depths were taken at six locations across the pond length and five locations across the pond width.

The average pond depth can be calculated as the average of all of these measurements. The volume of water in the pond in acre-feet is calculated by simply multiplying the pond area in acres by the average pond depth in feet.

Keep in mind that one acre-foot of water is equal togallons. An even better way to calculate an average pond depth is to divide the pond into numerous at least four sub-areas much like we did in the trapezoid method. Take at least one depth within each of the sub-areas and use these to calculate the overall average pond depth.

This method is especially good if the pond bottom is irregular rather than bowl shaped. Another important measurement for a pond or lake manager to know is the "residence time" of your water resource. Residence time is how long water resides in the pond, from the time it enters the pond to the time it leaves through the spillway or overflow pipe. This calculation is important for making herbicide or liming applications to the pond.In this rule, the boundaries between the ends of ordinates are assumed to be straight.

Thus the areas enclosed between the baseline and the irregular boundary lines are considered trapezoids.

The value of a definite integral is approximated using quadratic function. In this rule, the boundaries between the ends of ordinates are assumed to form an arc of parabola. Viva Differences. The result obtained by the trapezoidal rule is not affected because the boundary between the ordinates is considered straight. Trapezoidal rule can also be referred to as Trapezium rule.

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Gives an estimated result. For example, the area of an irregular piece of land obtained by trapezoidal rule is just an approximate value and not the accurate value.

There is no limitation for this rule. This rule can be applied for any number of ordinates. This rule is applicable only when the number of divisions is even i. Greater computational effort is involved and rounding errors may become a more significant problem.

## How to calculate the concrete quantity of a trapezoidal footing?

To the sum of the first and last ordinate, twice the sum of intermediate ordinate is added. This total sum is multiplied by the common distance. Half of this product is the required area. To the sum of the first and the last ordinate, four times the sum of even ordinates and twice the sum of the remaining odd ordinates are added.

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One third of this product is the required area.Why don't fictional characters say "goodbye" when they hang up a phone? All Rights Reserved. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. Hottest Questions. Previously Viewed. Unanswered Questions. Wiki User Related Questions Asked in Geometry What is the formula for volume of concrete footing in trapezoidal?

Asked in Geometry What is the formula to find volume of trapezoidal shape? If you mean volume of a trapezoidal prism then it is: 0. You cannot have a trapezoidal cylinder. That is like a square circle - an impossible shape.

Asked in Math and Arithmetic, Geometry How do you calculate the volume of concerete needed for trapezoidal footing? Asked in Math and Arithmetic, Volume What is the formula for calculating volume of eccentric footing?

Asked in Math and Arithmetic, Algebra, Geometry How do you find the volume of a trapezoidal pedestal? If it is a rectangular trapezoidal pedestal: Calculate the area of the base of the pedestal, multiply by the height of the pedestal. If the angles of the pedestal are not 90deg. Asked in Math and Arithmetic, Algebra, Geometry What is formula a trapezoidal cube with unequal bases? A cube cannot be trapezoidal.

A cube MUST have only square faces. Second, what is the formula supposed to tell you: the total surface area? Asked in Geometry, Algebra What is the formula for finding the volume of a trapezoidal prism? Area of trapezoidal cross-section x length. Multiply H by the area of the trapezoid that you found in step one. Asked in Math and Arithmetic, Geometry Formula for volume of a trapezoidal trough?

The answer depends on what you mean by a trapezoidal trough. Is it a trough whose cross-section is a rectangle at any height but which increases linearly with the height or is it a trough whose base and each face is a trapezium. Asked in Algebra, Geometry Formula for finding volume of a trapezoidal prism?

Asked in Geometry What is the volume of a trapazoidal prisam? It is the area of the trapezoidal face multiplied by the length of the prism. Asked in Math and Arithmetic How do you solve the volume of a triangular pyramid with a trapezoidal base? Asked in Geometry What is the volume of trapezoidal prism? It is: 0. Asked in Algebra, Geometry Volume of trapezoidal prism? To find the volume of a rectangular prism, multiply the base, width, and height.

If the prism is irregularly shaped, multiply the base and the height. Asked in Geometry What is the volume of a trapezoid prism? The volume of any right prism is the area of the base, in this case a trapezoid, multiplied by the height of the prism. Once you calculate the area of the trapezoidal base of the prism, multiply that number by its height to get its volume.This is sometimes called the Golenischev Papyrus, after the Russian who purchased it in Egypt in and brought it to Moscow, where it remains today.

It is also commonly called the Moscow Papyrus. It's about 18 feet centimeters long, and about 4 centimeters wide, and the writings consist of 25 mathematical problems with solutions. By far the most intriguing of these is the 14th, which asks for the volume of a truncated pyramid frustum.

Roughly translated, it says Given a truncated pyramid of height 6, base 4, and top 2, you are to square the bottom, multiply the bottom by the top, square the top, and add all these to give Then you are to multiply this by a third of the height to give the right answer, The type of solid described in this problem is illustrated below, with a, b, and h signifying the linear measures of the top, bottom, and height respectively.

In modern notation the Egyptian method of finding the volume can be expressed by the formula The level of sophistication of this result is quite a bit higher than that of the rest of the papyrus for example, the same papyrus gives incorrect formulas for some relatively simple plane areasleading some people to suspect that either the Egyptians just stumbled into this particular formula, or else perhaps it was part of a more advanced body of mathematical results not generally reflected in the papyrus.

Incidentally, according to tradition the god Thoth Djhowtryoriginally associated with time and the Moon, gave the calendar, astronomy, and mathematics "reckoning" to the Egyptians. Thoth was later identified with the Greek god Hermes, who was later called Hermes Trismegistos thrice greatthe supposed author of the hermetic works revealing the secret knowledge of the ancients.

Even as late as the 's this tradition was still influential in Europe.

## How To Calculate Trapezoidal Footing Volume

Isaac Newton, for example, was a devotee of hermetic studies, and actually seems to have believed that his own discoveries, such as calculus, universal gravitation, and much more, had been in the body of secret knowledge handed down from Thoth! Not surprisingly, Newton usually kept ideas like that to himself.

Obviously m is the linear dimension of the horizontal slice through the solid mid-way between the top and the bottom surfaces, and m 2 is the area of this mid-slice. Hence the Egyptian formula can be interpreted as giving the volume of any frustum in terms of the areas A 1A 2A 3 of the top, middle, and bottom slices This is interesting because it is identical to what is known in calculus as Simpson's Rule of integration.

It's easy to see why this is true for quadratic f xbecause it's essentially just the familiar integration rule for powers.

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Notice that this applied to truncated pyramids whose bases have any shape, not just square as drawn above. Hence the expression is sometimes called the prismoidal formula.

It is perhaps slightly less obvious that this same integration rule 1 applies to the general cubic polynomial f x as well. Similarly we can show that equation 1 applies to the general cubic polynomial. Not surprisingly, equation 1 is not exact for polynomials of degree greater than 3, but it is possible to devise analogous formulas applicable to higher degree polynomials in terms of the value of the polynomial at a sufficient number of intermediate values of x.In mathematicsand more specifically in numerical analysisthe trapezoidal rule also known as the trapezoid rule or trapezium rule is a technique for approximating the definite integral.

It follows that. The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sumsand is sometimes defined this way.

### Pond Measurements: Area, Volume and Residence Time

The integral can be even better approximated by partitioning the integration intervalapplying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" or "composite" trapezoidal rule is usually what is meant by "integrating with the trapezoidal rule".

When the partition has a regular spacing, as is often the case, the formula can be simplified for calculation efficiency. As discussed below, it is also possible to place error bounds on the accuracy of the value of a definite integral estimated using a trapezoidal rule.

A paper reports that the trapezoid rule was in use in Babylon before 50 BC for integrating the velocity of Jupiter along the ecliptic. The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result:. It follows that if the integrand is concave up and thus has a positive second derivativethen the error is negative and the trapezoidal rule overestimates the true value. This can also be seen from the geometric picture: the trapezoids include all of the area under the curve and extend over it.

Similarly, a concave-down function yields an underestimate because area is unaccounted for under the curve, but none is counted above. If the interval of the integral being approximated includes an inflection point, the error is harder to identify. Several techniques can be used to analyze the error, including: . It is argued that the speed of convergence of the trapezoidal rule reflects and can be used as a definition of classes of smoothness of the functions.

It then follows that. The trapezoidal rule converges rapidly for periodic functions. A similar effect is available for peak-like functions, such as GaussianExponentially modified Gaussian and other functions with derivatives at integration limits that can be neglected. Although some effort has been made to extend the Euler-Maclaurin summation formula to higher dimensions,  the most straightforward proof of the rapid convergence of the trapezoidal rule in higher dimensions is to reduce the problem to that of convergence of Fourier series.

For very large dimension, the shows that Monte-Carlo integration is most likely a better choice, but for 2 and 3 dimensions, equispaced sampling is efficient. This is exploited in computational solid state physics where equispaced sampling over primitive cells in the reciprocal lattice is known as Monkhorst-Pack integration.

For functions that are not in C 2the error bound given above is not applicable.If the number of sections is even, the end strip is treated separately and the area is calculated according to the trapezoidal rule. The volume of the remaining strips is calculated in the usual manner by the prismoidal formula.

Then both the results are added to obtain the total volume. The central heights at 40 m intervals are as follows: 0. Contour m. Area m 2. Login New User. Sign Up. Forgot Password? New User? Continue with Google Continue with Facebook. Gender Male Female. Create Account. Already Have an Account? Full Screen. All you need of Agricultural Engineering at this link: Agricultural Engineering. Download EduRev app here for Agricultural Engineering preparation.

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Content Details. Share with a friend. Share via Facebook. Share via Mail. Share via WhatsApp. It has gotten views and also has 4. Need something else? Download this Doc. Surveying and Levelling Notes- Agricultural Engg. Dynamic Test. Test this topic. Content Category. Continue with Google or Continue with Facebook. Why do I need to sign in? EduRev is a knowledge-sharing community that depends on everyone being able to pitch in when they know something.

What do I get? Follow Us.The length of the project along the centre line is divided into a series of solids known as prismoids by the planes of cross-sections.

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The spacing of the sections should depend upon the character of ground and the accuracy required in measurement. They are generally run at 20m or 30m intervals, but sections should also be taken at points of change from cutting to filling, if these are known, and at places where a marked change of slop occurs either longitudinally or transversely. The areas of the cross-sections which have been taken are first calculated and the volumes of the prismoids between successive cross- sections are then obtained by using the Trapezoidal formula or the prismoidal formula.

The former is used in the preliminary estimates and for ordinary results, while the latter is employed in the final estimates and for precise results. The prismoidal formula can be used directly or indirectly.

In the indirect method, the volume is firstly calculated by trapezoidal formula and the prisomoial correction is then applied to this volume so that the corrected volume is equal to that as if it has been calculated by applying the prisomoidal formula directly.

The indirect method being simpler is more commonly used. When the centre line of the project is curved in plan, the effect of curvature is also taken into account specially in final estimates of earthwork where much accuracy is needed. It is the common practice to calculate the volumes as straight as mentioned above and then to apply the correction for curvature to them.

Another method of finding curved volumes is to apply the correction for curvature to the areas of cross-sections, and then to compute the required volumes from the corrected areas from prismoidal formula.

The following are the various cross-sections usually met with whose areas are to be computed:. Formulae for the dimensions of the cross-sections of cutting and filling for the above cases are given below and should be verified by the readers as exercises. In this case, the ground is sloping transversely, but the slope of the ground does not intersect the formation level.

In this case the ground is sloping transversely, but the slope of the ground intersects the formation level such that one portion of the area is in cutting and the other in filling part cut and part fill.

When filling extends beyond the centre line i. In this case, the transverse slope of the ground is not uniform but-has multiple cross-slopes as is clear from the figure. The co-ordinates may be written in the determinant form irrespective of the signs. To calculate die volumes of the solids between sections, it must be assumed that they have some geometrical from.

They must nearly take the form of prismoids and therefore, in calculation work, they are considered to be prismoids. The number of cross-sections giving the areas may be odd or even. Since the areas at ends are averaged in this formula, therefore, it is also known as Average end Area formula. In order to apply the prismoidal formula, it is necessary to have odd number of sections giving the areas. If there be even areas, the prismoidal formula may be applied to odd number of areas and the volume between the last two sections may be obtained separately by trapezoidal formula and added.

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The difference between the volumes computed by the trapezoidal formula and the prismoidal formula is known as prismoidal correction. The volume by prismoidal formula is more nearly correct. Since the volume calculated by trapezoidal formula is usually more than that calculated by prismoidal formula, therefore the prismoidal correction is generally subtractive. In the formulae of prismoidal correction given below, the small and capital letters refer to the notations of the adjacent sections.

The prismoidal correction is denoted by C P.