Question

geometry***please help me find the surface area of this figure

Answer 1

We will determine the surface area of the cylinder as follows:

*First: We find the area of one of the bases [For each cylinder]:

**Bigger cylinder:

$A_{bC}=\pi(4)^2\Rightarrow A_{bC}=16\pi$

**Smaller cylinder:

$A_{bc}=\pi(1)^2\Rightarrow A_{bc}=\pi$

*Second: We determine the area of the face of the cylinder. For this, we can see the following:

When we "unroll" the face of the cylinder we will obtain a rectangle, and thus we find its area, that will be given by the circumference of the base of the cylinder times the length of the cylinder.

**Circumference bigger cylinder:

$C_{bc}=2\pi(4)\Rightarrow C_{bc}=8\pi$

**Circumference smaller cylinder:

$C_{sc}=2\pi(1)\Rightarrow C_{sc}=2\pi$

*Third: We find the area for each face of each cylinder:

**Area rectangle bigger cylinder:

$A_{rC}=8\pi\cdot6\Rightarrow A_{rC}=48\pi$

**Area rectangle smaller cylinder:

$A_{rc}=2\pi\cdot1\Rightarrow A_{rc}=2\pi$

*Fourt: We will determine the total surface area for the shape by adding the areas for the bigger cylinder and substract from it the surface area for the smaller cylinder, that is:

$T_s=(16\pi+48\pi)-(\pi+2\pi)\Rightarrow T_s=61\pi$

So, the total surface area for the hollow cylinder is 61pi square inches.