Area of interior triangle of pyramid normal to a side length

In summary, the conversation discusses the calculation of the area of a plane within a pyramid, normal to a specific point on the pyramid. The area is related to the volume of the pyramid and the angle of the taper. The problem is posed in two different ways, with one method leading to an inconsistent solution. Ultimately, the correct area can be calculated as one-third of the area of the base triangle.
  • #1
member 428835
Homework Statement
Given a pyramid who's base is an equilateral triangle with corner taper ##\psi##, what is the cross-sectional area of a plane that is normal to one of the interior corners (not normal to the altitude of the pyramid) ##z## distance from the corner taper?
Relevant Equations
A = 1/2 bh for a triangle
This isn't homework, but I figured it's fine if I make it a HW problem and post here (if not, please let me know).

Let ##z^*=0## be the vertex of the pyramid, and let ##z^*## run the altitude. It's easy to show the area of the base normal to the altitude is ##A = 4 \left.z^*\right.^2 \tan(30^\circ)\tan(\psi)^2##. However, if we now let ##z## define the length along one of the corner vertices, I'm not sure how to calculate the area of the plane within the pyramid normal to ##z##.

If we lay the pyramid on a side so that ##z## is going in the flat direction, I believe the taper angle would be ##2\psi## from ##z=0##, making me think the height of the interior triangle would be ##z\tan(2\psi)##, but I'm having issues thinking about that interior angle that used to be ##30^\circ##. Any help?

If this is confusing, let me pose the problem differently: given a pyramid with an equilateral triangular base ##ABC## and vertex ##V##, define ##O## to be the point on base ABC such that VO is the altitude (height) of the pyramid. ##\angle CVO= \psi##, the taper angle of the pyramid. Now let the point ##z## lie on ##\overleftrightarrow{VC}## such that ##\angle AzV = \angle BzV = 90^\circ##. What is the area of triangle ##AzB##?
 
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  • #2
joshmccraney said:
What is the area of triangle AzB?
Say it S
[tex]\frac{S\cdot VC}{3}=V[/tex]
where V is volume of the pyramid.
 
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  • #3
joshmccraney said:
Now let the point z lie on VC↔ such that ∠AzV=∠BzV=90∘. What is the area of triangle AzB?
Then z and O are the same point and the area of triangle AzB is 1/3 the area of triangle ABC.
these two conditions are inconsistent with the geometry of the problem you first stated
 
  • #4
I think I've been consistent with the notation, but in case not, here's a picture.
IMG_3761.jpg

Cool, so ##ABC \cdot VO = S\cdot Vz + S\cdot zC = S\cdot (Vz+zC) = S \cdot VC ##, which is three times the total volume. Thanks so much @anuttarasammyak !
 
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What is the formula for finding the area of the interior triangle of a pyramid normal to a side length?

The formula for finding the area of the interior triangle of a pyramid normal to a side length is A = (1/2)bh, where A is the area, b is the base length, and h is the height of the triangle.

Can the area of the interior triangle of a pyramid normal to a side length be negative?

No, the area of the interior triangle of a pyramid normal to a side length cannot be negative. It is always a positive value as it represents the amount of space enclosed by the triangle.

How is the height of the interior triangle of a pyramid normal to a side length calculated?

The height of the interior triangle of a pyramid normal to a side length can be calculated using the Pythagorean theorem, where the height is the hypotenuse and the base and side length are the other two sides.

Can the area of the interior triangle of a pyramid normal to a side length be greater than the area of the base of the pyramid?

Yes, the area of the interior triangle of a pyramid normal to a side length can be greater than the area of the base of the pyramid. This is because the base of the pyramid is a 2-dimensional shape, while the interior triangle is a 3-dimensional shape.

How is the area of the interior triangle of a pyramid normal to a side length related to the volume of the pyramid?

The area of the interior triangle of a pyramid normal to a side length is not directly related to the volume of the pyramid. However, it can be used to calculate the surface area of the pyramid, which is a factor in determining the volume.

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