Area of ​​cylinder: Surface area of ​​cylinder, total area of ​​cylinder

Let's learn about the formula for calculating the lateral area, total area and height of a cylinder to apply in study and daily life.

Table of Contents

• How to calculate the area of ​​a cylinder
  • Formula for calculating the lateral area of ​​a cylinder
  • Formula for calculating the total area of ​​a cylinder
• Calculate the height of the cylinder
• Formula for calculating the radius of the base of a cylinder
  • 1. Formula for calculating the circumference of a circle; area of ​​a circle 
  • 2. The base is the circle inscribed in the polygon
  • 3. The base is the circle circumscribing the polygon.
• What is a circular cylinder?
• Formula for calculating the cross-sectional area of ​​a cylinder 
• Example of calculating the area of ​​a cylinder 

How to calculate the area of ​​a cylinder

The area of ​​a cylinder includes the lateral area and the total area.

You can enter the height and radius of the cylinder into the table below to know the lateral area and total area of ​​the cylinder.

Formula for calculating the lateral area of ​​a cylinder

The lateral area of ​​a cylinder only includes the area of ​​the surrounding surface surrounding the cylinder, not including the area of ​​the two bases.

The formula for calculating the lateral area of ​​a cylinder is the circumference of the base circle multiplied by the height.

In there: Surrounding is the surrounding area. ris the radius of the cylinder. his the height, the distance between the two bases of the cylinder.

Example: 1

A circular cylinder has base radius r = 5 cm, height h = 7 cm. Calculate the lateral area of ​​the vertical cylinder.

Solution: Surface area of ​​a circular cylinder: Sxq = 2.π.rh = 2π.5.7 = 70π = 219.8 (cm2).

Example: 2

Given square ABCD with side length 2a. Let O and O' be the midpoints of sides AB and CD respectively. When rotating the square around axis OO', we get a cylinder of revolution. Calculate the lateral area of ​​the cylinder of revolution.

Solution:

The radius of the base circle is r= CD= a

The height of the cylinder is h= OO'= AD=2a

So the lateral area of ​​the cylinder is Sxq = 2πrh = 2π.a.2a =4a2π

Formula for calculating the total area of ​​a cylinder

The total area is calculated as the magnitude of the entire space occupied by the figure, including the lateral area and the area of ​​the two circular bases.

The formula for calculating the total area of ​​a cylinder is the lateral area plus the area of ​​the two bases.

Example 1 : Calculate the total area of ​​a cylinder with a base of 3 and a height of 5.

Solution:

The total area is Stp= Sxq + 2Sd = 2πr(r+h) = 2π.3(3+5) =48π

Calculate the height of the cylinder

The height of a cylinder is the distance between the two bases of the cylinder.

Calculate the height of a cylinder when knowing the total area and base radius

For example: Given a cylinder with base radius R = 8cm and total area 564π cm2 . Calculate the height of the cylinder.

Prize:

We have

Calculate the height of a cylinder when knowing the lateral area

=>

Formula for calculating the radius of the base of a cylinder

1. Formula for calculating the circumference of a circle; area of ​​a circle 

Circle has circumference C=2πr

=>

The circle with the base has area S=πr2

=>

Example. Calculate the radius of the cylinder base in the following cases:

a. The circumference of the base circle is 6π

b. The area of ​​the base is 25π

Solution:

a. The radius of the base circle is

b. The radius of the base circle is

2. The base is the circle inscribed in the polygon

- Inscribed in any triangle: with S being the area of ​​the triangle and p being the semi-perimeter

- Inscribed in an equilateral triangle: side

- Inscribed square:

Example 1. Given a cylinder inscribed in a cube with edge a. Calculate the radius of the cylinder.

The radius of the cylinder is:

Example 2. Given a regular prism ABC.A'B'C' with , the volume circumscribed around the cylinder. Calculate the radius of the cylinder.

The volume of the prism is

The base of a regular prism is an equilateral triangle, so => ​​the side

Therefore, the radius of the cylinder base is:

3. The base is the circle circumscribing the polygon.

Circumscribed in any triangle:

In there:

• a, b, c are the lengths of the three sides of the triangle
• p is the semiperimeter of the triangle:

Circumference of a right triangle: hypotenuse

Periphery of equilateral triangle: side

Circumference of square: side

For example: 

Calculate the base radius of the cylinder circumscribing the regular pyramid S.ABC in the following cases:

a. ABC is a right triangle at A with AB = a and AC = a√3

b. ABC has AB= 5; AC= 7; BC=8

Prize:

a. Hypotenuse

Because ABC is a right angle at A, radius R=0.5.BC=a

b. The semi-perimeter of triangle ABC is

What is a circular cylinder?

A circular cylinder is a cylinder with two equal circular bases parallel to each other.

Cylinders are used quite commonly in geometry problems from basic to complex, in which the formula for calculating the area and volume of a cylinder is often used. If you already know how to calculate the area and circumference of a circle, you can easily deduce the formulas for calculating the volume, lateral area as well as the total area of ​​a cylinder.

Formula for calculating the cross-sectional area of ​​a cylinder

Cut the cylinder by plane (P) through the axis

• The resulting cross section is a rectangle.

Cross-sectional area: SABCD = BC.CD =2r.h

Cut the cylinder by plane (P) parallel to and a distance x from the axis

The resulting cross-section is rectangle ABCD as shown above. Let H be the midpoint of CD, we have OH ⊥ CD=> Therefore the cross-sectional area

Cut the cylinder by plane (P) not perpendicular to the axis but cut all generators of the cylinder

The cross section formed is a circle with center O' and radius O'A'=r Cross-sectional area: S= πr2

Cut the cylinder by plane (P) not perpendicular to the axis but cut all generators of the cylinder.

The resulting cross-section is an Ellipse (E) with minor axis 2r => a=r Large axis equals with is the angle between the OI axis and (P) Therefore area S= π. ab=

Example of calculating the area of ​​a cylinder

Lesson 1 :

The lateral area of ​​a cylinder has a circular base circumference of 13cm and a height of 3cm.

Prize:

We have: circumference of circle C = 2R.π = 13cm, h = 3cm

So the lateral area of ​​the cylinder is:

Sxq = 2πr.h = Ch = 13.3 = 39 (cm²)

Exercise 2 : Given a cylinder with a base circle radius of 6cm, while the height from the base to the top of the cylinder is 8cm thick. What is the lateral area and total area of ​​the cylinder?

Prize

According to the formula, we have the base semicircle r = 6 cm and the height of the cylinder h = 8 cm. Therefore, we have the formula to calculate the lateral area of ​​the cylinder and the total area of ​​the cylinder as follows:

Surface area of ​​cylinder = 2 x π xrxh = 2 x π x 6 x 8 = ~ 301 cm²

Total area of ​​cylinder = 2 Π x R x (R + H) = 2 X π x 6 x (6 + 8) = ~ 527 cm²

Lesson 3 : A cylinder has a base radius of 7cm and a lateral area of ​​352cm2.

Then, the height of the cylinder is:

(A) 3.2 cm; (B) 4.6cm; (C) 1.8 cm

(D) 2.1cm; (E) Another result

Please select the correct answer.

Solution: We have

So, answer E is correct.

Lesson 4 : The height of a cylinder is equal to the radius of the base circle. The lateral area of ​​the cylinder is 314 cm2. Calculate the radius of the base circle and the volume of the cylinder (round the result to the second decimal place).

• Formula for calculating cylinder volume

Prize:

The lateral area of ​​the cylinder is 314cm2

We have Sxq = 2.π.rh = 314

Where r = h

So 2πr² = 314 => r² ≈ 50 => r ≈ 7.07 (cm)

Volume of cylinder: V = π.r2.h = π.r3 ≈ 1109.65 (cm³).

Hopefully the above article has helped you grasp basic and advanced knowledge about cylinders, how to calculate the total area and lateral area of ​​a cylinder.