Footing is one of the most fundamental and essential parts of the building structure.
It serves as the lowermost component of the structure and transfers the entire weight of the superstructure to the rock or soil below it.
Measuring the volume of footing is one of the crucial steps in the construction process.
It is done to estimate the quantity of materials like cement, sand, and aggregates required for the concrete used in the footing. It also helps in estimating the cost of constructing a footing before the actual construction process begins.
Calculating the volume of trapezoidal footing is somewhat difficult because we are not used to using these formulas in daily day-to-day activities.
So, In this article let’s discuss the formula used for calculating the volume of trapezoidal footing along with some examples.
Volume of trapezoidal footing formula
Isolated trapezoidal footing consists of two parts, namely
- A Cuboid and
- A Truncated Pyramid
The formula for calculating the volume of the cuboid is,
V1 = L x B x h
Where,
- L = Length of the footing
- B = Width of the footing
- h = Height of the cuboid
The formula for calculating the volume of the truncated pyramid is,
Where,
- H = Height of the truncated pyramid
- A1 = Area of the bottom base
- A2 = Area of the top base
So, The total volume of the trapezoidal footing is,
V = V1 + V2
Volume of Trapezoidal footing formula derivation
This section is provided only for those who want to know the derivation of the above formula. Now, Let’s discuss the derivation, shall we?
Let’s say, you take a pyramid and cut a portion of it from the apex. Now, the small portion of the pyramid that is removed is called the “Frustum of a pyramid” and the remaining part which has the original base and smaller top base is called the “Truncated (cutoff) pyramid”.
Let’s consider,
- A1 = Area of the bottom base of the pyramid
- A2 = Area of the top base of the truncated pyramid
- H = Height of the truncated pyramid
- h = Height of the whole pyramid
Volume of pyramid = 1/3 x Base area x Height of pyramid
Volume of pyramid = 1/3 x A1 x h — Eqn.1
Volume of the smaller pyramid which is to be removed = 1/3 x A2 x (h – H) — Eqn.2
Volume of truncated pyramid = Eqn.1 – Eqn.2
= (1/3 x A1 x h) – (1/3 x A2 x (h – H))
= (1/3 x A1 x h) – (1/3 x A2 x h) + (1/3 x A2 x H)
= (1/3 x (A1 – A2) x h) + (1/3 x A2 x H) — Eqn.3
Now, in the equation, we get 2 variables i.e., h and H. But for the truncated pyramid, we use the H variable. So let’s try to change the h variable into H.
Using the ratio to the area of bases and their heights, we get
Let’s substitute our h in eqn.3.
Now, take out 1/3 and H as the common factor. (A1 – A2) cancels each other. Then the remaining formula will be,
Example Problems
Example 1
1. Find out the volume of trapezoidal footing below.
i) Dimension of the cuboid:
- L=3m, B=3m, Thickness of cuboid=450mm (0.45m)
ii) Dimension of the truncated pyramid:
- Bottom dimensions of the truncated pyramid: L=3m, B=3m
- Top dimensions of the truncated pyramid: L=0.5m, B=0.5m
- Height of the truncated pyramid: H = 500mm (0.5m)
Calculation:
Volume of cuboid = 3 x 3 x 0.45 = 4.05 m3 (cubic meters) —1
Area of the bottom base of the truncated pyramid, A1 = 3 x 3 = 9m2
Area of the top base of the truncated pyramid, A2 = 0.5 x 0.5 = 0.25m2
Volume of truncated pyramid = (1/3) x 0.5 x (9 + 0.25 + Sqrt(9 x 0.25)) = 1.79 m3 —2
Total volume of trapezoidal footing = 1 + 2 = 4.05 + 1.79 = 5.84 m3
You can also check the result by using the online calculator mentioned below.
Example 2
2. Find out the volume of trapezoidal footing below.
i) Dimension of the cuboid:
- L=3m, B=2m, Thickness of cuboid=300mm (0.30m)
ii) Dimension of the truncated pyramid:
- Bottom dimensions of the truncated pyramid: L=3m, B=2m
- Top dimensions of the truncated pyramid: L=0.6m, B=0.5m
- Height of the truncated pyramid: H = 300mm (0.30m)
Calculation:
Volume of cuboid = 3 x 2 x 0.30 = 1.8 m3 (cubic meters) —1
Area of the bottom base of the truncated pyramid, A1 = 3 x 2 = 6m2
Area of the top base of the truncated pyramid, A2 = 0.6 x 0.5 = 0.30m2
Volume of truncated pyramid = (1/3) x 0.30 x (6 + 0.30 + Sqrt(6 x 0.30)) = 0.76 m3 —2
Total volume of trapezoidal footing = 1 + 2 = 1.8 + 0.76 = 2.56 m3
Example 3
3. Find out the volume of trapezoidal footing below.
i) Dimension of the cuboid:
- L=4m, B=4m, Thickness of cuboid=400mm (0.40m)
ii) Dimension of the truncated pyramid:
- Bottom dimensions of the truncated pyramid: L=3.5m, B=3.5m
- Top dimensions of the truncated pyramid: L=0.6m, B=0.6m
- Height of the truncated pyramid: H = 500mm (0.50m)
Calculation:
Volume of cuboid = 4 x 4 x 0.40 = 6.4 m3 (cubic meters) —1
Area of the bottom base of the truncated pyramid, A1 = 3.5 x 3.5 = 12.25m2
Area of the top base of the truncated pyramid, A2 = 0.6 x 0.6 = 0.36m2
Volume of truncated pyramid = (1/3) x 0.50 x (12.25 + 0.36 + Sqrt(12.25 x 0.36)) = 2.45 m3 —2
Total volume of trapezoidal footing = 1 + 2 = 6.4 + 2.45 = 8.85 m3
Example 4
4. Find out the volume of trapezoidal footing below.
i) Dimension of the cuboid:
- L=3m, B=3m, Thickness of cuboid=300mm (0.30m)
ii) Dimension of the truncated pyramid:
- Bottom dimensions of the truncated pyramid: Same as cuboid dimension
- Top dimensions of the truncated pyramid: Same as column dimension
- Height of the truncated pyramid: H = 450mm (0.45m)
iii) Column dimension:
- Length of column = 0.4m
- Width of column = 0.4m
Calculation:
Volume of cuboid = 3 x 3 x 0.30 = 2.7 m3 (cubic meters) —1
Area of the bottom base of the truncated pyramid, A1 = 3 x 3 = 9m2
Area of the top base of the truncated pyramid, A2 = 0.4 x 0.4 = 0.16m2
Volume of truncated pyramid = (1/3) x 0.45 x (9 + 0.16 + Sqrt(9 x 0.16)) = 1.55 m3 —2
Total volume of trapezoidal footing = 1 + 2 = 2.7 + 1.55 = 4.25 m3
Volume of Trapezoidal Footing Online Calculator
Volume of Trapezoidal Footing Online Calculator
Cuboid volume
Truncated pyramid volume
Result:
Volume of cuboid =
Volume of truncated pyramid =
Total volume of footing =
Key takeaways
Hope you understand everything you need to know about calculating the volume of trapezoidal footing. If you have queries or if you found this article helpful, let us know in the comment section.
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