Great Pyramid of Giza

Geometry Level 4

The image above shows the Great Pyramid of Giza, which can be regarded as a square pyramid.

If a square whose side length is the height of the pyramid has the same area as one of the triangles of the pyramid, what is the ratio of the height of the one of the triangles of the pyramid to the side length of the bottom square of the pyramid?

Source: Gaokao 2020, I

5 − 1 4 \dfrac{\sqrt{5}-1}{4} 5 + 1 4 \dfrac{\sqrt{5}+1}{4} 5 − 1 2 \dfrac{\sqrt{5}-1}{2} 5 + 1 2 \dfrac{\sqrt{5}+1}{2}

Alice Smith by Alice Smith , 20, China

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1 solution

Chris Lewis
Jul 29, 2020

Let the side of the square base be b b , the height of the pyramid h h , and the height of one of the triangles s s ; let x = s b x=\frac{s}{b} be the ratio we're after.

b , h , s b,h,s are related by h 2 = s 2 − b 2 4 h^2=s^2-\frac{b^2}{4} .

The area of one of the triangular faces is 1 2 b s \frac12 bs , so from the given condition, h 2 = 1 2 b s s 2 − b 2 4 = 1 2 b s 4 s 2 − b 2 = 2 b s 4 x 2 − 1 = 2 x 4 x 2 − 2 x − 1 = 0 \begin{aligned} h^2&=\frac12 bs \\ s^2-\frac{b^2}{4}&= \frac12 bs \\ 4s^2-b^2 &=2bs \\ 4x^2-1 &=2x \\ 4x^2-2x-1 &= 0 \end{aligned}

The solutions of this quadratic are x = 1 ± 5 4 x=\frac{1\pm \sqrt{5}}{4} . We of course want the positive root; so the answer is x = 1 + 5 4 x=\boxed{\frac{1+ \sqrt{5}}{4}} .

3 Helpful 1 Interesting 0 Brilliant 0 Confused

What is x x ?

Shubhrajit Sadhukhan - 4 months, 1 week ago

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x x is defined in the first line.

Chris Lewis - 4 months, 1 week ago

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