Formula for calculating the surface area of a cylinder

Among geometric shapes, a cylinder is relatively simple, but remembering how to calculate the surface area of a cylinder can be quite tricky. Let's explore what the calculation formula is all about!

Formula for calculating the surface area of a cylinder

General formula for calculating the surface area of a cylinder

- Surface area of the cylinder's lateral side:

SAS = 2πrh

+ In simple terms: The lateral surface area of a cylinder equals twice the product of the cylinder's radius and height, multiplied by pi.

- Total surface area of the cylinder:

S_tsa = S_las + S_{two bases} = 2πrh + 2πr²

+ In simple terms: The total surface area of a cylinder equals the sum of its lateral surface area and the area of its two bases.

- Notation: S_las, S_tsa: Are respectively the lateral surface area and the total surface area of the cylinder.

S_{two bases}: The area of the two bases of the cylinder
r: Radius of the circular base
h: Height
- Measurement unit: square meters (m²)

Application exercises:

1. Calculate the surface area of the cylinder including lateral surface area and total surface area, given:

a) r = 5 cm, h = 12 cm
b) r = 3.3 dm, h = 5.1 dm
c) r = 6/7 m, h = 3/2 m
d) r = 10 cm, h = 23 cm

2. Calculate the total surface area of a cylinder with a base circumference of 30 cm and a height of 6 cm.

3. A cylinder has a lateral surface area of 418 cm2, and the radius of its base is 14 cm.

a) Find the height of the cylinder
b) Calculate the total surface area of the cylinder.

* Exercise-solving tips: For these exercises, you all use the formulas for the lateral surface area and total surface area of a cylinder to solve them.

Additional knowledge alongside the formula for the surface area of a cylinder

1. Distinguishing between the concepts of lateral surface, cylinder, and solid cylinder

- Lateral surface (also known as the curved surface): It's a circle formed when a fixed line d rotates around a movable line d', always parallel to d and at a distance equal to R.

+ d' is the axis
+ R is the radius
+ d is the generating line

=> Additionally: The lateral surface can also be understood as the set of all points at a fixed distance R.

- Cylinder: It's a shape bounded by the lateral surface and two equal circles, which are the intersection of the lateral surface with two planes perpendicular to the axis. In other words, a cylinder is formed by rotating a rectangle around one of its sides.

+ The two bases are two equal circles parallel to each other, each lying on a different plane.

- Solid cylinder: It's the cylinder along with its interior.

2. Cylinders inscribed and circumscribed with spheres

- When the base of the cylinder is two circles on the surface of a sphere (S), then the cylinder T is called inscribed in the sphere (S).
- When the axis of the cylinder is the diameter of the sphere (S), then the cylinder T' with radius R and height 2R is called circumscribed about the sphere (S).

We hope the knowledge we provide here will be beneficial to our readers, especially students, in solving geometry exercises related to calculating the surface area of cylinders. You can also refer to formulas for calculating the volume of cylinders, how to calculate the area of rectangles, squares, etc., in our other articles.

Students also need to review and master how to calculate the area of circles in plane geometry. This is fundamental knowledge that students need to remember to avoid difficulties when dealing with problems related to circles.