Diagonal length is all we need

Geometry Level 1

What is the surface area of a cube with a 10-meter space diagonal?

Give your answer in square meters.


The answer is 200.

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A Former Brilliant Member by A Former Brilliant Member

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3 solutions

From the figure above, considering one face diagonal (by Pythagorean theorem )

y = x 2 + x 2 = 2 x 2 = x 2 y = \sqrt{x^2 + x^2} = \sqrt{2x^2} = x\sqrt{2}

Considering the space diagonal, we have

1 0 2 = x 2 + ( x 2 ) 2 10^2 = x^2 + (x\sqrt{2})^2

x = 10 3 x = \frac{10}{\sqrt{3}}

Surface area = 6 x 2 = 6 ( 10 3 ) 2 = 200 6x^2 = 6(\frac{10}{\sqrt{3}})^2 =\boxed{ 200} square meters.

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Moderator note:

This solution demonstrates a good circumstance to avoid approximating at intermediate steps. With rounding to the second decimal point x x comes out to be 5.77 , 5.77, x 2 x^2 comes out to be 33.41 , 33.41, and 6 x 2 6x^2 is 200.46. 200.46. This is far enough from the actual answer of 200 200 square meters that it could easily be a source of error.

Also note the solution avoids rationalizing the intermediate equation of x = 10 3 x = \frac{10}{\sqrt{3}} into x = 10 3 3 , x = \frac{10\sqrt{3}}{3}, which here would be an unnecessary extra step.

x x itself isn't really needed to be honest. x 2 x^2 is enough since you need that 6 times.

Peter van der Linden - 4 years, 4 months ago

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yeah , you are right.. shorter solution

A Former Brilliant Member - 4 years, 4 months ago

That's a 3-4-5 right triangle: Pythagorean theorem. The hypotenuse is 2(5)=10 and the short leg is x= 2(3)=6, and the other leg y= 2(4)=8. x is a side of the cube. 6x6 =36 gives the area of one face of the cube, and the cube has 6 sides, so 36x6= 216, the total surface area of the cube. Or simply 6x squared, or 6x6x6=216, or am I missing something?

Meghan Lefferts - 4 years ago

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Can you explain to [email protected]. I think you are wrong

Janesh G - 3 years, 8 months ago

You have the correct idea except the 8m now becomes the hypotenuse for the bottom facing square and we can assume the bottom facing as a right triangle as well because its a cube. So 8/5 is 1.6m Hypotenuse. 1.6 3=4.8m and 1.6 4=6.4m. The equation should look something like this 2(6x4.8)+2(6x6.4)+2(4.8x6.4)= 195.84

Peter Panagopoulos - 3 years, 2 months ago

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This comment section messed up my math in the comment for some reason

The bottom facing square is 4.8m x 6.4m. As seen in the third sentence of my last comment

Peter Panagopoulos - 3 years, 2 months ago

Peter, except for the bottom face you shouldn't use the 3-4-5 triple, since it should be an isosceles triangle... Giving a side length of 8 2 \frac8{\sqrt2} or 5.657, which doesn't equal the height of 6. So the object is not a cube.

Meghan, you can't assume the diagonal section of the cube is similar to the notorious 3-4-5... since it actually has to be 1-1.4142-1.732.

C . - 2 years, 9 months ago

if x is 5.77, then x^2 should be 33.29... giving a result of 199.74. I can't help but wonder how you could get an excess when you rounded down?!

C . - 2 years, 9 months ago
Vu Vincent
Feb 2, 2017

X X = one side of cube The formula for the cube diagonal is ( 3 X 2 ) = 10 \sqrt(3X^{2}) = 10

S A = 2 ( X 2 + X 2 + X 2 ) = 6 X 2 SA = 2(X^2+X^2+X^2) = 6X^2

We know that:

3 X 2 = 100 3X^2 = 100

⇒ 6 X 2 = 200 \Rightarrow 6X^2 = 200

Therefore the surface area is 200 m 200\text{ m}

12 Helpful 0 Interesting 0 Brilliant 0 Confused

You could use LaTeX to format the equations. I believe your √3x^2 is meant to be 3 x 2 \sqrt{3x^2} but it looks more like 3 x 2 \sqrt{3}x^2 . Otherwise, nice way to find the answer without explicitly solving for x x !

Christopher Boo - 4 years, 4 months ago

Is it just me I did it the easy way. A=6Xsquared so the diagonal would be A=2Dsquared which is 10x10=100 and 100x2=200

Michelle Pimentel - 4 years, 1 month ago

the answer is not 200 but 200m^2 HHAAAHA!!

Syed Hamza Khalid - 4 years, 3 months ago

We know that the diagonal of a cube is 3^{½}a and surface area of a cube is 6a² where a is the side length of the cube. So, in this question- a = 10/3^{½} and 6a² = 6. 10²/(3^{½})² . = 200

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Could you explain how do you get the diagonal of a cube is 3 a \sqrt{3} a ?

Christopher Boo - 4 years, 3 months ago

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In a cube with side length a, in any surface the length of the diagonal is 2^{½}a.(by applying Pythagorean theorem) Again the height of the cube is also a and if we form a triangle with the diagonal and height(like in the pic above) it'll be a right angle triangle. So, by applying Pythagorean theorem we will get- (2^{½}a)² + a² = (diagonal of the cube)² then, diagonal = 3^{½}a

Omar Sayeed Saimum - 4 years, 3 months ago

@Omar Sayeed Saimum @Christopher Boo we can use the direct formula to calculate the surface area surface area of the cube= 2.d.d, where d=diagonal length of the cube. Therefore required surface area=2.10.10=200 square meters

Venkatachalam J - 4 years, 3 months ago

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I got 203.646752981725687

Abe Alaniz - 2 years, 11 months ago

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explain how you got the number 203.646752981725687. Answer is 200 square meters

Venkatachalam J - 2 years, 11 months ago

improve more

improve more - 2 years, 9 months ago

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