Blue Or Red?

Let $h$ denote the height of the blue cylinder. Note that the blue cylinder of has a height equivalent to the height of the 2 hemispheres, $h = r + r = 2r$ .

Recall that the total surface area of a cylinder is equal to the sum of areas of the two base circles (one on top, one at the bottom) and the curved rectangular side. Expressing it mathematically, we have

$2(\pi r^2) + 2\pi r h = 2 \pi r^2 + 2\pi r (2r) = 6\pi r^2 \qquad \qquad (1)$

The total surface area of the two hemispheres in pink is equal to the sum of curved surface area and the sum of the two base circles (one on top, one at the bottom). And the curved surface area of each of these hemisphere is simply half the surface area of a sphere , $4\pi r^2$ . Expressing all of these mathematically, we have

$2 \left ( \dfrac12 \cdot 4 \pi r^2 \right) + 2 ( \pi r^2) = 6 \pi r^2 \qquad \qquad \qquad \qquad (2)$

Since the values we've found in $(1)$ and $(2)$ are equal, then both the blue figure and the pink figure has equal surface area.

Computation of the surface area of the cylinder:

$h=2r$

$A=lateral~area+area~of~the~two~bases=(2 \pi r)(2r)+2(\pi r^2)=4 \pi r^2+2 \pi r^2=6 \pi r^2$

Computation of the surface area of the two hemisphere:

$A=2(2 \pi r^2) + 2(\pi r^2)=4 \pi r^2 + 2 \pi r^2=6 \pi r^2$

Conclusion: The surface areas are equal.

Both have equal circular surfaces at the ends that can be disregarded.

The height is 2r, dictated by the configuration of the two hemispheres. (Same deal as the chocolate sphere in the volume quiz)

The area of the cylinder side is that of a rectangle 2r x 2πr (that is height x circumference), that is, $4πr^2$ as if the side were simply cut at a right angle then rolled out flat.

There are two hemispheres with the total area of one one sphere, $4πr^2$ (formula from Archimedes) .

Thus they are the same.

Cylinder:

assume r = 1

from the figure, h = 2r = 2(1) = 2

A = area of the two bases + lateral area = 2 pi r² + 2 pi r*h = 2pi + 4pi = 6pi

Two Hemispheres:

A = 2 x area of one hemisphere = 2(2 pi r² + pi*r²) = 2(2pi + pi) = 4pi + 2pi = 6pi

Conclusion: The two areas are equal.

the surface area is = to the area of the cross section x the height. both shapes have the same cross sectional area ( area of the the circle ) and the same height which leads to an = surface area

The top surfaces are both the same shape and size and the radius is the same

The height of blue cylinder is 2r and now calculate both of their surface areas.You will get 6πr^2 as their total surface area