answersLogoWhite

0


Best Answer

(1)

Best formula to use is as follows -

V = h/3(Areatop + √(Areatop*Areabottom) + Areabottom)

(2)

To find (h) using a tape measure -

Areatop => bd

Areabase => ac

Lateral edge remaining => e (from top corner to base corner)

k = 1 - √(bd/ac)

H= √([e/k]² - [a/2]² - [c/2]²)

h = Hk

V = H/3*(ac-bd+bd*k)

(3)

Lets say Top is a rectangle with sides b & d

and bottom is a rectangle with sides a & c respectively.

Let height be h

in that case the volume of Truncated Pyramid with rectangular base will be -

V = 1/3((a²c-b²d)/(a-b))h

BUT BE CAREFUL - a,b,c,d are not all independent variables (one depends on the others) so this answer is misleading!!!

Proof -

Suppose the height of Full Pyramid is H

From parallel line property

(H-h)/H = b/a

Rearranging

H = ah/(a-b) --------------------(1)

Also

Since V=1/3 Base area X Height

Volume of full pyramid = 1/3 X ac X H

Volume of removed Pyramid = 1/3 X bd X (H-h)

So volume of truncated part V = 1/3(acH-bd(H-h))

=1/3((ac-bd)H + bdh)

From (1)

V = 1/3((ac-bd)ah/(a-b) + bdh)

reducing and rearranging we get

V = 1/3((a²c-b²d)/(a-b))h

(4)

In case the truncated solid forms a prism instead, we have following formula -

V = ( h/6)(ad + bc + 2ac + 2bd)

Proof -

Fig(1)

Fig(2)

Lets divide the fig(1) into four different shapes as shown in fig(2)

VA = Volume of cuboid = bdh

VB = Volume of prism after joining both Bs= ½ X base X height X width = ½ (a-b) (d)(h)

VC = Volume of prism after joining both Cs = ½ X base X height X width = ½ (c-d) (b)(h)

VD = Volume of rectangular pyramid after joining all Ds = 1/3 X base area X height =

1/3 (a-b) (c-d) h

Then V = VA + VB + VC + VD

Or, V = bdh + 1/2(a-b)dh +1/2(c-d)bh + 1/3(a-b)(c-d)h

Arranging and simplifying we get -

V = ( h/6)(ad + bc + 2ac + 2bd)

User Avatar

Wiki User

10y ago
This answer is:
User Avatar

Add your answer:

Earn +20 pts
Q: Volume of a truncated rectangular based pyramid?
Write your answer...
Submit
Still have questions?
magnify glass
imp
Related questions

What is the given volume of a rectangular pyramid?

The volume of any pyramid or cone is given by the formula: 1/3 x base area x height For a rectangular-based pyramid: 1/3 x base width x base length x height


Rectangular based pyramid?

I don't really understand your question, but one example of a rectangular based pyramid is the Great Pyramid in Giza.


What is a rectangular based pyramid with the top cut off?

It is a frustum of a rectangular pyramid.


What is a figure with four faces that are triangles and rectangle base?

In trigonometry, we call it a rectangular pyramid.


What is a rectangular based not square pyramid called?

A rectangle based pyramid!


What 3d shape has 7 sides?

A heptahedron.There are 34 distinct heptahedra: some of these are:a hexagon based pyramid,a pentagon based pyramid with one of the basal vertices truncated,a cuboid with one of its vertices truncated.


Which solid has one rectangular face?

a rectangular based pyramid


How many vertices does a rectangular pris have than rectangular pyramid?

A rectangular prism has 8 vertices whereas rectangular based pyramid has 5 vertices


How many vertices of a rectangular pyramid have?

A rectangular based pyramid has 5 vertices, 8 edges and 5 faces


How many faces has a triangular based prism pyramid have?

five A triangular based prism has 5 A triangular based pyramid has 4 and there is no such thing as a triangular based prism pyramid - unless you mean a truncated pyramid (or a frustum) which is essentially a prism.


How many vertices's edges and faces has a rectangular based pyramid got?

a rectangular based pyramid has got 6 edges and four faces and four vertices.


What does a truncated octahedron look like?

You start with a quadrilateral based bipyramid, which has six vertices. At each vertex, remove the tip in the form of a quadrilateral based pyramid. You will be left with a truncated octahedron.