Pythagorean Geometry

Geometry Level pending

Based on the dimensions of this pyramid, what is the volume (in centimeters cubed) of the smallest rectangular prism in which this polyhedron can fit?

Note: The four edges from the base to the apical vertex are each 13 c m cm in length.


The answer is 576.

Liam P by Liam P , 22, Canada

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1 solution

let x x = slant height

h h = height

Solving for the slant height

Using Pythagorean Theorem

x = 1 3 2 − 4 2 = 153 x = \sqrt{13^2 - 4^2} = \sqrt{153} c m cm

Solving for the height

Using Pythagorean Theorem

h = x 2 − 3 2 = ( 153 ) 2 − 3 2 = 153 − 9 = 144 = 12 h = \sqrt{x^2 - 3^2}= \sqrt{(\sqrt{153})^2 - 3^2} = \sqrt{153 - 9} = \sqrt{144} = 12 c m cm

Solving for the volume of the rectangular prism

V = A b a s e h = ( 8 ) ( 6 ) ( 12 ) = V = A_{base}h = (8)(6)(12) = 576 c m 3 \boxed{576cm^3}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

From the image it is not clear that all the side edges are length 13. (Since the two sides of the base are different from one another, symmetry is not heavily implied.) It should be either shown or stated that it is so.

Marta Reece - 4 years, 2 months ago

While that is the most obvious "rectangular prism", how do you know it gives us the smallest volume? E.g. Why can't I fit the prism in a slightly tilted angle?

Calvin Lin Staff - 4 years, 2 months ago

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