Dome, Ball, Plate

Geometry Level 2

A small green circle is inscribed within the section of a bigger blue circle, touching the mid-chord, as shown above left. Then the graphs are revolved around the y y -axis to generate three figures: a blue cover dome, a green spherical melon, and a red serving plate.

Which of the following options will have more surface area?

I. The blue dome
II. The melon plus the plate

The melon plus the plate Both options have the same surface area Not enough information The blue dome

View wiki
Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Let R R be the radius of the big circle; r r be the radius of the red plate; and h h be the diameter of the green sphere.

It is obvious that the red circular area = π r 2 \pi r^2 and the surface area of the green sphere = 4 π ( h 2 ) 2 = π h 2 4\pi (\dfrac{h}{2})^2 =\pi h^2

Now in order to calculate the spherical section area of the blue dome, we need to find out the radius of the original full sphere, R R in terms of r r and h h :

According to the diagram above, by using Pythagorean theorem, R 2 = ( R − h ) 2 + r 2 R^2 = (R-h)^2 + r^2 .

Thus, 2 R h = h 2 + r 2 2Rh = h^2 + r^2 .

Hence, R = h 2 + r 2 2 h R = \dfrac{h^2 + r^2}{2h} .

Now according to Archimedes' Hat-Box Theorem , any spherical section from spherical radius R R and of height h h will have its lateral surface area equal to the lateral surface area of a cylinder of radius R R and height h h :

That is, the spherical section area = 2 π R h = π ( 2 h ) h 2 + r 2 2 h = π ( h 2 + r 2 ) 2\pi Rh = \pi (2h)\dfrac{h^2 + r^2}{2h} = \pi(h^2 + r^2) .

Thus, clearly the surface area of the blue dome = surface area of a green sphere + red circular plate.

4 Helpful 0 Interesting 0 Brilliant 2 Confused

Instead of "it is obvious" wouldn't it be better to just say "by the area and surface area formulas"? I actually had to parse for a bit to decide if there was some logical step being passed over (which is usually what that phrase indicates).

Jason Dyer Staff - 4 years, 6 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...