SURFACE AREA, VOLUME AND CAPACITY (Cuboid, Cube and Cylinder)
202. Cylinder Formulas Unrolled · CSA, TSA, and Volume from first principles
The formulas for CSA, TSA, and Volume are consistent with the dimensions r and h.
Cylinder formulas (radius r, height h): Volume = πr²h (base area × height), CSA = 2πrh (curved surface, the "label"), TSA = 2πrh + 2πr² (CSA + two circular caps). All three update as you change r or h.
What this lesson covers
Try to break it
Drag r to change the radius and h to change the height. Base area πr², curved surface 2πrh, and total surface 2πr(r + h) all update together. Push r toward zero and the whole cylinder vanishes — every formula politely goes to 0 with it.
How you build it
Construct the net of a cylinder.
- Draw the curved surface as a rectangle — width 2πr (the base circumference) by height h.
- Attach a base circle of radius r at the left end of the rectangle.
- Attach the second base circle of radius r at the right end to complete the net.
The proof, step by step
Prove that the CSA, TSA, and volume formulas of a cylinder match its radius r and height h.
- The area of the base circle is πr².
- The circumference of the base is 2πr.
- The curved surface unrolls into a rectangle of width 2πr and height h.
- Curved Surface Area = 2πr × h.
- Total Surface Area = Curved Surface Area + 2 × Base Area = 2πrh + 2πr² = 2πr(h + r).
- Volume = Base Area × Height = πr²h.
Worked example
A cylindrical tank has a radius of 7 cm and a height of 10 cm. Find its curved surface area and volume. (Use π = 22/7)
CSA = 2πrh = 2 × 22/7 × 7 × 10 = 440 cm². Volume = πr²h = 22/7 × 49 × 10 = 1540 cm³.
- CSA = 440 cm², Volume = 1540 cm³ — correct
- CSA = 220 cm², Volume = 770 cm³
- CSA = 440 cm², Volume = 770 cm³
- CSA = 220 cm², Volume = 1540 cm³