Volume of pyramid with 2 toppoints

I know how to calculate a pyramid's volume when I have the base and perpendicular height to the 1 toppoint. It's by

$$\frac13 \cdot \mbox{base} \cdot \mbox{height}$$

However, what if the top is not a point, but a line? Is this easy to calculate?

Also, is there a difference between an pyramid with a regular rectangle as base, and a pyramid with some other shaped base (for example a irregular plane with 6 points.). Cause at the end I need to calculate it for a pyramid with an irregular base. But, at first, an rectangle as base would be a good start.

edit: for clearification: line E-F is parallel to plan ABCD:

for clearification: line E-F is parallel to plane ABCD and line CD:

the second one can be splitted in a prisma and a piramid, but how about the first one?

• @Goodqustion. Your question is not clear and it will also be closed in here. You have to ask first for a pyramid with a rectangle base and a line on the top. You may get an answer. Commented Apr 24, 2021 at 11:30
• why? The formula 1/3*base*height does also count for ALL base shapes right? not just a rectangle... but ok... i will change it if you like. Commented Apr 24, 2021 at 12:02
• You mean a roof shape with 2 identical trapezoids and two identical isosceles triangles ? Commented Apr 24, 2021 at 12:18
• yeah when talking about a regular rectangle as base you might say that. However, I didn't say the topline is parallel to a baseline. but would that matter? guess not? Commented Apr 24, 2021 at 12:27
• There are two immediate ways forward: $(1)$ break up your figure into a variety of easily calculable areas, and add them up, or $(2)$ find a way to parameterize the figure in $\mathbb{R}^3$ and integrate the volume. Commented Apr 24, 2021 at 12:33

A simple form

In figure top of the pyramid is a line , lower height is h and higher height is k. Base is rectangle , sides a and b.Suppose the coordinates of lower top is $$(0, \frac b2, h$$ and that of higher top is $$(a, \frac b2, k)$$; The slope of the line on the top is:

$$m=\frac{k-h}a$$

So the equation of the line on the top is:

Or: $$z=\frac{k-h}a x+h$$

The volume of pyramid is:

$$V=\int^b_0dy\int^a_0 dx\int^{z=\frac{k-h}x x+h}_0 dz=\frac {k-h}a a^2b+abh$$

Example: a=4, b=2, k=3, h=2:

$$V=\frac{3-2}4\times 4^2\times 2+4\times 2\times 2= 24$$

If you put $$h=0$$ the figure becomes a pyramid and it's volume is:

$$V=\frac14\times 4^2\times 2=8$$

We compare his with when top is a point:

$$v=\frac 13 \times 4\times 2=\frac 83$$

If the line on top is parallel with zy plane at hight h, then there will be a triangle on side(on ZY plane) it;s area is $$A=\frac 12 bh$$ and the volume of pyramid is $$V= A.b=\frac12 abh$$. For example a=4, b=2 and h=3 $$\rightarrow V=\frac12 \cdot 4\cdot 2\cdot3=12$$. If base is irregular but we know its area and also the legth of the line on the top, then we can find volume as:

$$b=\frac Aa$$

where A is area of base, a legth of the line on the top and b is equivalent width of the base if rectangular . So volume is:

$$V=\frac12 ab h =\frac 12 Ah$$

The volume of figure in your question is:

$$V= A_{ABCD}\times\frac{h_{AED}+h_{BFC}}2$$

where $$A_{ABCD}$$ is the area of base ABCD and $$h_{AED}$$ and $$h_{BFC}$$ are the heights of walls AED and BFC.

• Thanks for your answer, but I can't fully follow it. Next to that I don't know if the image is wrong or if i'm interpreting it wrong, so added a picture to my first post. However. Your first step: what is z-0? and x-0? I don't understand this. Are you calculation the length of the topline by the ratio??? Commented May 7, 2021 at 19:55
• I edited my answer and changed the form of presentation. Hope it helps. Commented May 8, 2021 at 5:48
• Don't know what you see as height, the height of the top (point E or F) perpendicular to the surface ABCD? Or the height of the top (point E or F) to the baseline of the triangle (line AB or line BC). However, I drawn the figure in Civil, with base 2 x 2, and height of line EF perp to base ABCD 2, and get an result different from your fomula. When doing Aabcd = 2x2 = 4, and h is smallest = (2+2)/2, this gives 4 x 2 = 8 which would be volume of the cube, it's easy to see the volume should be less then that. Commented May 11, 2021 at 13:53
• Yes the walls are supposed to be perpendicular on surface. If they are not then you have 3 pyramids joined; one in the middle with vertical walls and two other with wall having slope with surface.In this case you must have the slopes. Commented May 11, 2021 at 16:20