In this chapter we will discuss the concept of hollow cylinder and learn about different formulas for calculating their volume and surface area.
What is hollow cylinder ?
A cylinder which is empty from inside but has thickness that forms internal and external radius of cylinder.
One can imagine hollow cylinder as a cylinder inside the bigger cylinder.
Given below is the image of hollow cylinder.
The radius of internal cylinder is r.
Radius of external cylinder is given by R.
In hollow cylinder, the area between inside and outside cylinder is not empty and is filled with material.
Volume of material of hollow cylinder
Here we are basically calculating the volume of material that’s present between inside and outside cylinder.
The formula is given as;
Volume of material = \mathtt{\ \pi \ \left( R^{2} -r^{2}\right) \ h}
Curved surface area of hollow cylinder
Now in hollow cylinder, there are two different curved surface; internal and external curved surface.
The formulas is given as;
Inner curved surface area = 2.𝜋.r.h
Outer curved surface area = 2.𝜋.R.h
Total surface area of hollow cylinder
Here the total surface area consists of internal and external curved surface and surface area of top and base rim.
TSA = 2.𝜋.r.h + 2.𝜋.R.h + 2.𝜋. \mathtt{\left( R^{2} -r^{2}\right)}
Please memorize the above formulas as we will use it for solving questions.
Questions on hollow cylinder
Question 01
Given is the hollow cylinder with external radius 8 cm and height 10 cm. If the total surface area of cylinder is 338𝜋 sq. cm, then find the internal radius of cylinder. Note that the hollow cylinder is open at both sides.
Solution
External radius (R) = 8 cm
Height (h) = 10 cm
Total surface area = 338𝜋 sq. cm
Internal radius (r) = ?
Formula for total surface area;
TSA = 2.𝜋.r.h + 2.𝜋.R.h + 2.𝜋. \mathtt{\left( R^{2} -r^{2}\right)}
Putting the values;
\mathtt{338\pi =2\pi h( r+R) +2\pi \left( R^{2} -r^{2}\right)}\\\ \\ \mathtt{169=10( r+8) +\left( 64-r^{2}\right)}\\\ \\ \mathtt{169\ =\ 10r+80+64-r^{2}}\\\ \\ \mathtt{r^{2} -10r+25=0}\\\ \\ \mathtt{( r-5)^{2} =0}\\\ \\ \mathtt{r=5\ cm\ }
Hence, the internal radius of hollow cylinder is 5 cm .
Question 02
The internal radius of hollow iron pipe is 12 cm and outer radius is 14 cm. If the heigh of hollow cylinder is 50 cm then find the mass of the pipe. 1 cu cm of iron weigh 7 grams.
Solution
Finding the volume of iron mass of hollow cylinder.
Volume of material = \mathtt{\ \pi \ \left( R^{2} -r^{2}\right) \ h}
Putting values in the formula;
\mathtt{Volume\ =\frac{22}{7}\left( 14^{2} -12^{2}\right) \times 50}\\\ \\ \mathtt{Volume\ =\ \frac{22}{7}( 52) \times 50=\frac{57200}{7} cm^{3}}
So 57200/7 cu cm is the volume of hollow pipe.
Now let’s calculate the mass of the cylinder.
1 cu cm ⟹ 7 grams
57200/7 cu cm ⟹ 57200/7 x 7 = 57200 grams
Hence, the mass of hollow cylinder is 57200 grams