I cut off one corner of an empty box to form an open 'pocket', so that each of the length a, breadth b, and depth c, of the initial corner was a different whole number of centimetres, and the area of each face of the 'pocket' was also an integer.
What is the area of the open 4th side of the 'pocket', in terms of a,b, and c?
The information about integer side lengths may have been part of a separate question, but is irrelevant to determining the formula for the area of the open 4th side of the box in terms of a,b,c.
I went through the process of calculating the area of the 4th side triangle using Heron's formula. Considering the corner as the origin, the vertices of the triangle are at (a,0,0), (0,b,0) and (0,0,c).
The sides of the triangle are sqrt(a^2+b^2), sqrt(a^2+c^2), sqrt(b^2+c^2)
The semiperimeter, s, is [sqrt(a^2+b^2) + sqrt(a^2+c^2) + sqrt(b^2+c^2)] / 2
And after much algebra, all the a^4, b^4 and c^4 terms drop out; and all the a^3b^1 type terms drop out as well. And finally:
A = (1/2)* sqrt(a^2b^2 + a^2c^2 + b^2c^2)
Edited on December 19, 2020, 1:13 pm
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Posted by Larry
on 2020-12-19 13:11:39 |
explanation
