design of a hat

Geometry Level pending

A little boy is going to a party tomorrow. He wants to have a weird hat. So, from a cardboard, he constructed a hat (figure $A$ ). The height is 60 cm and the base is in the shape of a regular hexagon with an edge length of 9 cm. He then cuts the upper portion of the hat by passing a perpendicular plane 42 cm from the base to have the final design (figure $B$ ). What is the surface area (in square centimeters) of the final design? Give your answer to the nearest integer.

Remarks:

Figure A is in the shape of a regular pyramid.
Don't include the area inside the hat.

The answer is 1487.

1 solution

A Former Brilliant Member
Jul 16, 2017

By similar triangles, we have

$\dfrac{x}{42}=\dfrac{9}{60}$ $\color{#3D99F6}\large \implies$ $x=6.3$

$y=\sqrt{42^2+6.3^2}=\sqrt{1803.69}$

$z=9-x$ $\large \implies$ $z=9-6.3=2.7$

$a=\dfrac{9-2.7}{2}=3.15$

$L=\sqrt{\left(\sqrt{1803.69}\right)^2-3.15^2} \approx 42.353$

Since the figure is in the shape of a frustum of pyramid, the surface area is given by

$S=\dfrac{1}{2}(p+P)(L)$ where $p$ is the perimeter of the upper base, $P$ is the perimeter of the lower base and $L$ is the slant height

So, we have

$S=\dfrac{1}{2}[6(2.7)+6(9)](42.353) \approx$ $\color{#D61F06}\large \boxed{1487}$