is the angle between the OI axis and (P)
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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
- Calculate the height of the cylinder
- Formula for calculating the radius of the base of a cylinder
- 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.
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In there:
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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.
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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.
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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.
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Cross-sectional area: SABCD = BC.CD =2r.h |
Cut the cylinder by plane (P) parallel to and a distance x from the axis
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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
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Cut the cylinder by plane (P) not perpendicular to the axis but cut all generators of the cylinder
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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.
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The resulting cross-section is an Ellipse (E) with minor axis 2r => a=r Large axis equals with 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).
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.
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