Geometry or Algebra?

Geometry Level 2

The sum of the lengths of all edges of a rectangular box is 140 cm, and the distance from one vertex of the box to its opposite verrtex is 21 cm. What is the total surface area (in $\text{cm}^{2}$ ) of the box?

The answer is 784.

1 solution

Michael Fuller
May 6, 2015

Let the three lengths of the box be $a$ , $b$ and $c$ . Then $4a+4b+4c=140\\ a+b+c=35$ Using 3D Pythagoras, the length from corner to corner of the box is $\sqrt { { a }^{ 2 }+{ b }^{ 2 }+c^{ 2 } } =21\\ { a }^{ 2 }+{ b }^{ 2 }+c^{ 2 }=441$ Now, ${ (a+b+c) }^{ 2 }={ a }^{ 2 }+{ b }^{ 2 }+c^{ 2 }+2(ab+bc+ca)\\ 1225=441+2(ab+bc+ca)\\ 2(ab+bc+ca)=\large \color{#20A900}{\boxed { 784 }}$

Note that the sum of lengths is 140, and not 40.

Calvin Lin Staff - 5 years, 7 months ago

I have no idea what was going on with the numerical values there haha. Thanks.

Michael Fuller - 5 years, 7 months ago

Geometry or Algebra?

The answer is 784.

1 solution

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