Calculation of Area and Volume MCQ Quiz - Objective Question with Answer for Calculation of Area and Volume - Download Free PDF
Last updated on Apr 28, 2025
Latest Calculation of Area and Volume MCQ Objective Questions
Calculation of Area and Volume Question 1:
If the ordinates are provided as 25 m, 44 m, and 20 m, then the value of d is 2 m. Determine the area using the mid-ordinate.
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 1 Detailed Solution
Concept:
The formula for the area of the mid-ordinate can be given as
Area = common distance × sum of mid-ordinates
Area of plot = h1 × d + h2 × d + … + hn × d = d (h1 + h2 + … hn)
A = d × ∑O.
Calculation:
Given that,
d = 2 m
Ordinate = 25 m, 44 m, and 20 m
On substitution,
A = 2 × (25 + 44 + 20)
A = 178 sq. m.
Calculation of Area and Volume Question 2:
The sum of first and last ordinates, add twice the sum of the remaining odd ordinates and four times the sum of all the even ordinates. The total sum thus obtained is multiplied by one-third of the common distance between the ordinates and the result gives the required area. This rule of finding the area is called:
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 2 Detailed Solution
Explanation:
Simpson's rule:
This rule is based on the assumption that the figures are trapezoids.
In order to apply Simpson's rule, the area must be divided in even number i.e., the number of offsets must be odd i.e., n term in the last offset 'On' should be odd.
The area is given by Simpson's rule:
\(Area = \frac{d}{3}\left[ {({O_1} + {O_n}) + 4({O_2} + {O_4} + ........ + {O_{n - 1}}) + 2({O_3} + {O_5} + ......{O_{n - 2}})} \right]\)
where O1, O2, O3, .........On is the offset
Important Points
- In case of an even number of cross-sections, the end strip is treated separately and the area of the remaining strip is calculated by Simpson's rule. The area of the last strip can be calculated by either trapezoidal or Simpson's rule.
Calculation of Area and Volume Question 3:
Estimate the quantity of earthwork for a portion of road using the trapezoidal formula, in which distance between the sections of the road D=30 m and cross-sectional areas \(A_0 = 5.5 \, \text{m}^2, \quad A_1 = 11.5 \, \text{m}^2, \quad A_2 = 17 \, \text{m}^2, \quad A_3 = 24.5 \, \text{m}^2 \)
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 3 Detailed Solution
Concept:
To estimate the earthwork quantity between multiple cross-sections, we use the Trapezoidal Formula:
\( V = D \left[ \frac{A_0 + A_n}{2} + A_1 + A_2 + \dots + A_{n-1} \right] \)
Given:
Distance between sections, \( D = 30 \, \text{m} \)
Cross-sectional areas: \( A_0 = 5.5 \, \text{m}^2, \, A_1 = 11.5 \, \text{m}^2, \, A_2 = 17 \, \text{m}^2, \, A_3 = 24.5 \, \text{m}^2 \)
Calculation:
Apply the trapezoidal formula:
\(V = 30 \left[ \frac{5.5 + 24.5}{2} + 11.5 + 17 \right]\)
\(V = 30 \left[ \frac{30}{2} + 11.5 + 17 \right] = 30 \left[15 + 11.5 + 17 \right]\)
\(V = 30 \times 43.5 = 1305 \, \text{m}^3\)
Calculation of Area and Volume Question 4:
The trapezoidal formula is used to calculate ______.
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 4 Detailed Solution
Explanation:
Trapezoidal Formula Application
The trapezoidal formula is used to calculate the area of irregular plots. It is particularly useful in scenarios where the plot does not have a regular shape, and the boundaries are not straight lines. By dividing the irregular plot into several trapezoids, we can calculate the area of each trapezoid and then sum them to get the total area. This method is widely used in land surveying and civil engineering to accurately determine the area of land plots.
Analyzing the Given Options
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"Volume of excavation." (Incorrect)
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The volume of excavation is usually calculated using other methods such as the cross-sectional area method or the prismoidal formula, not the trapezoidal formula.
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The trapezoidal formula focuses on area calculation, not volume.
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"Area of irregular plots." (Correct)
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The trapezoidal formula is specifically designed to calculate the area of irregularly shaped plots by dividing them into a series of trapezoids.
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Summing the areas of these trapezoids gives the total area of the plot, making it the correct application of the trapezoidal formula.
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"Quantity of bricks." (Incorrect)
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The quantity of bricks is generally calculated using the dimensions of the wall or structure and the size of individual bricks, not using the trapezoidal formula.
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The trapezoidal formula does not apply to the calculation of brick quantities as it is intended for area calculations.
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"Reinforcement steel weight." (Incorrect)
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The weight of reinforcement steel is typically calculated by determining the volume and density of the steel, not by using the trapezoidal formula.
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While the trapezoidal formula is useful for area calculations, it does not apply to the weight calculations of reinforcement steel.
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Calculation of Area and Volume Question 5:
If B is the width of formation, d is the height of the embankment, side slope is S : 1, for a highway with no transverse slope, then the area of cross-section is given by
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 5 Detailed Solution
Explanation:
Given,
B = Formation width
d = Height of the embankment
S:1 = Side slope of the embankment (H: V)
The area of embankment = The area of the trapezium
Area of Trapezium = \(\frac{1}{2}\) × (Sum of Lengths of parallel sides) × Height
Area of Embankment \(= \frac{1}{2}\left[ {B + \left( {B + 2sd} \right)} \right] \times d\)
A = Bd + sd2Top Calculation of Area and Volume MCQ Objective Questions
Prismoidal correction, while surveying is always?
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 6 Detailed Solution
Download Solution PDFExplanation:
The volume of earthwork by trapezoidal method = V1
V1 = \(common\: distance\left \{ \frac{First\: area + Last\: area}{2}+the \:sum \:of\: remaining \:area \right \}\)
The volume of earthwork by prismoidal formula = V2
V2 = \(=\frac{Common\: distance}{3}\left \{ First\: area+ Last\: area + 2(Sum\: of odd\: area) + 4(Sum\: of even\: area)\right \}\)
Prismoidal correction:
- The volume by the prismoidal formula is more accurate than any other method
- But the trapezoidal method is more often used for calculating the volume of earthwork in the field.
- The difference between the volume computed by the trapezoidal formula and the prismoidal formula is known as a prismoidal correction.
- Since the trapezoidal formula always overestimates the volume, the prismoidal correction is always subtractive in nature is usually more than calculated by the prismoidal formula, therefore the prismoidal correction is generally subtractive.
- Volume by prismoidal formula = volume by the trapezoidal formula - prismoidal correction
Prismoidal correction (CP)
\(C_{P}=\frac{DS}{6}\left \{ d- d_{1}\right \}^{2}\)
Where, D = Distance between the sections, S (Horizontal) : 1 (Vertical) = Side slope, d and d1 are the depth of earthwork at the centerline
In earthwork computations on a longitudinal profile, the diagram prepared to work out the quantity of earthwork is:
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 7 Detailed Solution
Download Solution PDFExplanation:
Mass Haul Curve:
- This is a curve representing the cumulative volume of earthwork at any point on the curve, the manner in which earth to be removed.
- It is necessary to plan the movement of excavated soil of worksite from cuts to fill so that haul distance is minimum to reduce the cost of earthwork.
- The mass haul diagram helps to determine the economy in a better way.
- The mass haul diagram is a curve plotted on a distance base with the ordinate at any point on the curve representing the algebraic sum of the volume of earthwork up that point.
A haul refers to the transportation of your project’s excavated materials. The haul includes the movement of material from the position where you excavated it to the disposal area or a specified location. A haul is also sometimes referred to as an authorized haul.
Haul = Σ Volume of earthwork × Distance moved.
What is the volume of earthwork for constructing a tank that is excavated in the level ground to a depth of 4 m ? The top of the tank is rectangular in shape having an area of 50 m × 40 m and the side slope of the tank is 2: 1 (horizontal: vertical).
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 8 Detailed Solution
Download Solution PDFConcept:
a) Trapezoidal Formula:
Volume (v) of earthwork between a number of sections having areas A1, A2,…, An spaced at a constant distance d.
\({\rm{V}} = {\rm{d}}\left[ {\frac{{{{\rm{A}}_1} + {{\rm{A}}_{\rm{n}}}}}{2} + {{\rm{A}}_2} + {{\rm{A}}_3} + \ldots + {{\rm{A}}_{{\rm{n}} - 1{\rm{\;}}}}} \right]\)
Also can be written as
\(V = \frac{D}{2}\left[ {({A_1} + A_n +2 \times (A2 +..+ A_{n-1} )} \right]\)
Calculation:
Given, A1 = 50 × 40 = 2000 m2, As the side slope is given 2:1 i.e. H:V
So for a depth of 1 m, there is a change of 2 m in a Horizontal Direction.
So at 4 m vertical depth
Bottom Dimension is ( 50 - 16 ) = 34 m & (40 - 16) = 24 m
∴ The bottom area is 34 m × 24 m = 816 m2
Mean area (Am) = (2000 + 816) / 2 = 1408 m2
According to simple Trapezoidal rule for volume,
\(V = \frac{D}{2}\left[ {({A_1} + 2Am + {A_2})} \right]\)
∴\(V = \frac{2}{2}\left[ {(2000 + 2\times (1408) + 816)} \right]\) = 5632 m3
The Simpson’s rule for determination of areas is used when the number of offsets are:
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 9 Detailed Solution
Download Solution PDFExplanation:
Simpson's rule:
This rule is based on the assumption that the figures are trapezoids.
In order to apply Simpson's rule, the area must be divided in even number i.e., the number of offsets must be odd i.e., n term in the last offset 'On' should be odd.
The area is given by Simpson's rule:
\(Area = \frac{d}{3}\left[ {({O_1} + {O_n}) + 4({O_2} + {O_4} + ........ + {O_{n - 1}}) + 2({O_3} + {O_5} + ......{O_{n - 2}})} \right]\)
where O1, O2, O3, .........On is the offset
Important Points
- In case of an even number of cross-sections, the end strip is treated separately and the area of the remaining strip is calculated by Simpson's rule. The area of the last strip can be calculated by either trapezoidal or Simpson's rule.
To the sum of the first and last ordinates, add twice the sum of the intermediate ordinates. The total sum thus obtained is multiplied by the common distance between the ordinates. One-half of this product gives the required area. This rule of finding the area is called
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 10 Detailed Solution
Download Solution PDFConcept:
i) Mid-Ordinate Rule
Area of plot = h1 × d + h2 × d + … + hn × d = d (h1 + h2 + … hn)
∴ Area = common distance × sum of mid-ordinates
ii) Average-Ordinate Rule
\(\text{Area}=\frac{{{\text{O}}_{1}}+{{\text{O}}_{2}}+\ldots +{{\text{O}}_{\text{n}}}}{{{\text{O}}_{\text{n}+1}}}\times \text{l}\)
\(\text{i}.\text{e}.\text{ }\!\!~\!\!\text{ Area}=\frac{\text{Sum }\;\!\!~\!\!\text{ of }\;\!\!~\!\!\text{ ordinates}}{\text{No}.\;\text{ }\!\!~\!\!\text{ of }\;\!\!~\!\!\text{ ordinates}}\times \text{length }\;\!\!~\!\!\text{ of }\;\!\!~\!\!\text{ base }\;\!\!~\!\!\text{ line}\)
(iii) Trapezoidal Rule
\(\text{Total }{area}=\frac{\text{d}}{2}\left\{ {{\text{O}}_{1}}+2{{\text{O}}_{1}}+2{{\text{O}}_{2}}+\ldots +2{{\text{O}}_{\text{n}-1}}+{{\text{O}}_{\text{n}}} \right\}\)
\(\text{Total Area}=\frac{\text{Common distance}}{2}\times \{\left( 1\text{st ordinate}+\text{last ordinate} \right)+2\left( \text{sum of other ordinate} \right))\}\)
(iv) Simpson’s Rule
\(\text{Total }{ area}=\frac{\text{d}}{3}\left( {{\text{O}}_{1}}+4{{\text{O}}_{2}}+2{{\text{O}}_{3}}+4{{\text{O}}_{4}}+\ldots +{{\text{O}}_{\text{n}}} \right)\)
\(\text{Total Area}=\frac{\text{Common Distance}}{3}\times \left\{ \left( 1\text{st ordinate}+\text{last ordinate} \right)+4\left( \text{sum of even ordinates} \right)+2\left( \text{sum of remaining odd ordinates} \right) \right\}\)
Which of the following formulas explains the calculation of area of earthwork using its mean depth?
Area = BD + SD2
Where, B = Breadth of section
D = Mean depth of section
SD = Areas of sides
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 11 Detailed Solution
Download Solution PDFExplanation:
Methods for measurement of earthwork:
1. Mid Section method 2. Trapezoidal Method 3. Prismoidal Method 4. Simpson's 3/8th rule
Mid Section Method:
In this method the quantity of earthwork is computed with the help of size of mid section.
Volume of earthwork \(= Area\times Length=\left\{BD+SD^{2} \right\} \times L \)
Determine the approximate quantity of earthwork for a road in embankment having a length of 120 m on a uniform level ground. The width of formation is 10 m and side slopes are 3 ∶ 1. The heights of the bank at the ends are 1 m and 1.5 m, respectively. Use trapezoidal method considering average of areas at the two ends.
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 12 Detailed Solution
Download Solution PDFConcept:
Volume of Sloped earthwork (V) = (bd + sd2) × L
Where, B = Width, d = Depth and S = Side slope of the cross-section
Calculation:
Given,
slope = s: 1 = 3: 1
b = 10 m
L = 120 m
d1 = 1.5 m and d2 = 1 m
Volume of Sloped earthwork (V) = (bd + sd2) × L
\(V = \;\frac{1}{2}\left[ {(b{d_1}\; + {\rm{ }}s{d_1}^2) + (b{d_2}\; + {\rm{ }}s{d_2}^2)} \right] \times 120\)
\(V = \;\frac{1}{2}\left[ {(10 \times 1.5\; + {\rm{ }}3 \times {1.5^2}) + (10 \times 1\; + {\rm{ }}3 \times {{1}^2})} \right] \times 120 = 2085{m^3}\)
The approximate quantity of earthwork = 2085 m3
A road embankment 10 m wide at the formation level with side slopes 2:1 and with an average height of 5 m is constructed with an average gradient of 1:40 from the contour 220 m to 280 m. Find the volume of earthwork.
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 13 Detailed Solution
Download Solution PDFConcept:
Gradient:
A gradient is the rate of rise or falls along the length of the road with respect to horizontal. It is expressed as ‘1' vertical unit to 'N' horizontal units.
\(Tan\ \theta \ =\frac{{\bf{h}}}{{\bf{l}}}\)
\(\frac{{\bf{h}}}{{\bf{l}}} = \frac{1}{{N}}\)
Area of trapezoidal:
According to the trapezoid area formula, the area of a trapezoid is equal to half the product of the height and the sum of the two bases.
Area = ½ x (Sum of parallel sides) x (perpendicular distance between the parallel sides).
Calculation:
Road embankment = 10 m
Average height = 5 m
Difference in elevation(h) = 280 - 220 = 60 m
Average gradient = \( \frac{1}{{40}}\)
\(\frac{{\bf{h}}}{{\bf{l}}} = \frac{1}{{40}}\)
\(\frac{{\bf{60}}}{{\bf{l}}} = \frac{1}{{40}}\)
L = 2400 m
Average cross-sectional area(A) = \(% MathType!MTEF!2!1!+- % feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaWcaaWdaeaapeGaaGymaaWdaeaapeGaaGOmaaaacqGHxdaTdaqa % daWdaeaapeGaaGymaiaaicdacqGHRaWkcaaIZaGaaGimaaGaayjkai % aawMcaaiabgEna0kaaiwdaaaa!423F! \frac{1}{2} \times \left( {10 + 30} \right) \times 5\)\(\frac{1}{2} \times \left( {10 + 30} \right) \times 5\)
A = 100 m2
Volume of earthwork = A × L
Volume of earthwork = 100 × 2400
∴ Volume of earthwork = 2,40,000 m3
What is the volume of a 6 m deep tank having rectangular shaped top 6 m x 4 m and bottom 4 m x 2m (computed through the use of prismoidal formula)?
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 14 Detailed Solution
Download Solution PDFConcept:
Calculation:
Given: L = 6m
Top area = 6 × 4 = 24 m2 = A1
Bottom area = 4 × 2 = 8 m2 = A2
\({{\rm{A}}_{\rm{m}}} = \left( {\frac{{6 + 4}}{2}} \right) \times \left( {\frac{{4 + 2}}{2}} \right) = 15{\rm{\;}}{{\rm{m}}^2}\)
\({\rm{Volume\;}} = \frac{{\rm{L}}}{6}\left( {{{\rm{A}}_1} + 4{{\rm{A}}_{\rm{m}}} + {{\rm{A}}_2}} \right)\)
\({\rm{Volume}} = \frac{6}{6}\left( {24 + 4 \times 15 + 8} \right) = 92{\rm{\;}}{{\rm{m}}^3}\)
To apply Simpson’s rule for computation of irregular area, number of segments should be-
Answer (Detailed Solution Below)
Calculation of Area and Volume Question 15 Detailed Solution
Download Solution PDFExplanation: Simpson's rule:
- Simpson's rule is also known as Parabolic Rule.
- For calculation of volume, this formula is known as the Prismoidal rule
- This rule is used when the ends of ordinates or straight to form an arc of a parabola.
- This rule is applicable only when the number of divisions is even, i.e. the number of ordinates is odd.
- This rule can be stated as follows: One-third of common distance multiplied by the sum of last and first ordinate, four times the sum of even ordinates, and twice the sum of the remaining odd ordinates are added.
Additional Information
Simson's Rule | Trapezoidal Rule |
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The boundary between the ordinates is considered to be an arc of a parabola. | The boundary between the ordinates is considered to be straight. |
It gives a more accurate result. |
It gives an approximate result |