Square-Based Pyramid Volume

Lektrix · 2003-02-20T15:50:48

For my G.C.S.E. coursework, I have decided to try and find the largest possible (within reason) volume of certain shapes when made of a net with an area of 100 cm². The size of the shape is dependent the size of an angle on the net. I have done a few and developed a base class and some subtypes for the shapes I have done. I am stuck on square-based pyramids though. I have some working formulae, yet given some sizes of degrees it does not work (I shall come onto this in a minute). Here is a quick image of what the net of my square-based pyramid looks like (please excuse, I am terrible at creating images ). net.gif. A is the size of the angle which is changed to try and find the largest volume possible. 25 cm² is the area of each of the net triangles (100 / 4 = 25). The problem that I have is that I need the pyramid's slant height, in order to use pythag to work the pyramid's height out. When one folds out the pyramid to form the net, the net triangles' heights are the pyramid's slant heights. Please look at the following image of one net triangle: net_triangle.gif. l = net triangle's height & pyramid's slant height x = half of the net triangle's base A = size of angle Area of net triangle = 25 cm² I figured that I could work out l by rearranging the formula for finding the area of a triangle (half base * height = area): x * l / 2 = 12.5 (12.5 is half of the net triangle's area). But there would be an unknown left in the formula, x. I used trig to find the formula for finding the value of x and substituted it in place of x in the formula for calculating the area of the net triangle: *gets breath back* tan = opp / adj tan(½A) = x / l x = tan(½A) * l x * l / 2 = 12.5 => tan(½A) * l * l / 2 = 12.5 Now I can rearrange the formula, making l to subject: tan(½A) * l² / 2 = 12.5 tan(½A) * l² = 25 l² = 25 / tan(½A) l = sqrt(25 / tan(½A)) This is ok for some values of A, but if the tan of half of A is a negative number it makes the whole formula not work: 25 divided by a negative number will result in a negative number and there is no such thing as a square root of a negative number. So, how can I get over this? I know this is a school project, so I am not saying that you should explicitly state what to do, yet some help would be nice. Someone mentioned maybe complex numbers for the square root of a negative number? Anyway, thanks in advance. Edit: Messed up URLs :0 [ Google || Start Here || ACCU || MSDN || STL || GameCoding || BarrysWorld || E-Mail Me ] [edited by - Lektrix on February 20, 2003 9:37:19 AM]

Math and Physics Programming

Started by Lektrix February 20, 2003 08:31 AM

11 comments, last by Lektrix 22 years ago

Lektrix

Author

106

February 20, 2003 01:41 PM

Oh, thank you very much. I was unaware of that. I shall search on Google for a conversion rate.

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grhodes_at_work

1,385

February 20, 2003 02:06 PM

quote:
Original post by Lektrix
Oh, thank you very much. I was unaware of that. I shall search on Google for a conversion rate.

Conversion is simple. Since there are 2*PI radians in 360 degrees, or PI radians in 180 degrees:

radians = degrees * PI/180

PI = 3.14159265 of course,

or

PI = 4.*atan(1.)

I''d actually use a precalculated factor:

DEG_TO_RAD = 4.*atan(1.)/180.

then

radian = degrees * DEG_TO_RAD

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

Graham Rhodes Moderator, Math & Physics forum @ gamedev.net

Keem

138

February 20, 2003 03:50 PM

The surface area should include the base of the pyramid. Saying it doesn''t count towards the surface area is like saying the lenght of the base of a square doesn''t count towards it perimeter. Check with your teacher to see exactly what he wants.

Also, the best way to solve this type of problem is by calculus. I know that it''s not on the GCSE syllabus but it you bug your teacher I''m sure they''ll show you what to do ( i did a similar problem for GCSE, finding the maximum volume of a cube from a given rectangle of paper, showing the basics of calculus will get you that ellusive A* on the course work mark!!!)

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