Math Is Fun Forum

Identity

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Registered: 2007-04-18

Posts: 934

Angle between two faces

In a pyramid OABCD with square base length a and height h. How can you find the angle between two adjacent triangular faces?
thanks

John E. Franklin

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Registered: 2005-08-29

Posts: 3,588

Re: Angle between two faces

Extremes:
The tallest pyramid: angle approaches 90 degrees.
The shortest pyramid: angle approaches 180 degrees.

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igloo myrtilles fourmis

JaneFairfax

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Registered: 2007-02-23

Posts: 6,868

Re: Angle between two faces

Is the answer

?

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Angle between two faces

Is the answer

?

Yes! Jane that is correct. Could you roughly tell me how you got it?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Angle between two faces

Calculate the angle between the two planes.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Angle between two faces

I am getting a possibly equivalent equation to Jane!! Yeah!!
But the formula looks different, but if you try
h=10 and a=6, or h=22 and a=55, then
we both come out to 94.73623625 degrees and 127.5718693 degrees.
Wow!! Looks like they might be the same!!
I can't believe I did this, I'm new to this subject.

Last edited by John E. Franklin (2008-08-02 12:35:55)

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igloo myrtilles fourmis

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: Angle between two faces

Do I construct midpoints on each of the sloping edges and join them together to form a triangle? Seems a bit too complicated for the kind of problem the book would give.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Angle between two faces

If you want to create 2 lines that bend around the midpoint of an edge,
you could do it that way somehow.  The 2 lines will probably have to
be orthogonal to the edge that goes up between the 2 sides.  Each line
will be drawn like with a pencil on the pyramid.
I'm assuming given 0ABCD, that O is the top, and ABCD is the square
going around the base.  So 4 triangles already exist:
triangles: OAB, OBC, OCD, and ODA.

Last edited by John E. Franklin (2008-08-02 17:44:47)

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igloo myrtilles fourmis

JaneFairfax

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Registered: 2007-02-23

Posts: 6,868

Re: Angle between two faces

John, first calculate the length of the sloping edge (the edge between two adjacent triangular faces) of the pyramid. Call this length s. Can you do that? The answer should be

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Angle between two faces

Yes, your "s" is correct.

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igloo myrtilles fourmis

JaneFairfax

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Registered: 2007-02-23

Posts: 6,868

Re: Angle between two faces

Good. Next step, take a triangular face, and draw the perpendicular from the vertex to the midpoint of the base. Calculate the length p of this perpendicular. It should be

So p is slightly shorter than s.

Let M be the midpoint of the base. Now draw MN perpendicular to one of the sloping edges, and let d = |MN|. Using similar triangles, verify that

Last edited by JaneFairfax (2008-08-03 13:22:50)

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Angle between two faces

Yes!!  I see! Because a/2 is the hypotenuse of the small right triangle,
and "s" is the hypotenuse of the big right triangle.  It is very
interesting to note that the two small right triangles only look
like the big one, if you look from the backside at it.  Thank
you for sharing this with us.

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igloo myrtilles fourmis

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Angle between two faces

You’re welcome.

Repeat the procedure with the triangular face adjacent to the sloping edge containing the point N. Draw the perpendicular from the vertex to the midpoint L of the base; then LN is perpendicular to the sloping edge and |LN| = d.

And angle MNL is the angle between the two faces; it’s what you need to calculate. Consider triangle MNL. You have |MN| = |LN| = d and |ML| =

. So calculate ∠MNL.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Angle between two faces

Nice! Very Nice.  You turned a 3-D problem into
an isosceles triangle.   And we know that an
isosceles triangle can be cut vertically in this case
to make 2 right triangles back to back.   So
angle between planes is 2 invsin(|ML|/2d).

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igloo myrtilles fourmis

supramolecular

Member

Registered: 2009-05-09

Posts: 1

Re: Angle between two faces

Hi Jane, can you please define exactly where N is in this diagram?

You’re welcome.

Repeat the procedure with the triangular face adjacent to the sloping edge containing the point N. Draw the perpendicular from the vertex to the midpoint L of the base; then LN is perpendicular to the sloping edge and |LN| = d.

And angle MNL is the angle between the two faces; it’s what you need to calculate. Consider triangle MNL. You have |MN| = |LN| = d and |ML| =

. So calculate ∠MNL.

I'd appreciate it if you could draw a diagram.

Thanks.

whatismath

Member

Registered: 2007-04-10

Posts: 19

Re: Angle between two faces

I would like to show a diagram an essentially equivalent
evaluation of the angle.

The angle between two planes is the angle between two
intersecting lines respectively in the planes perpendicular
to the line of intersection of the planes. In the figure, OA
is the line of intersection of planes OAB and OAD. By
symmetry, the perpendiculars dropped from B and D
on OA intersect, say at point E.

is the angle between planes
OAB and OAD. We see

is
isosceles with BD=

. So we need
to find BE.

Consider

. BE is the altitude
if OA is the base. Let F be the midpoint of AB. Alternatively,
OF is the altitude if AB is the base. Thus the area of

has 2 expressions:

.
It is straigtforward to find

and

.
So

and

.
Then

.

Last edited by whatismath (2009-05-18 21:32:25)