Cylinder

The cylinder is a three-dimensional shape consisting of two bases (two identical circles, parallel to each other) and a lateral surface located in the middle between these two bases that surrounds and connects them.

To better illustrate the definition, you may be impressed by the following figure:

Cylinder Characteristics

  • The cylinder is bounded by two circles and the intersection connecting them.
  • The two circles overlap and are parallel to each other (since they are three-dimensional, this means that the two planes in which the circles are also parallel to each other).
  • Each of the circles is called a base.
  • The distance between the bases is fixed and is called the height of the cylinder (in the drawings H H , it is actually also the width of the rectangle in the second drawing).
  • The radius of each of the two circles is defined as R R
  • Each cylinder is defined by its radius (R) (R) and height (H) (H) .

Before addressing the geometrical characteristics of the cylinder, we will first clarify a number of important concepts related to the shape of the cylinder.

In this figure, you can see a cylinder, while in the drawing on the next line, you can get an impression of the arrangement of that cylinder, if we were to disassemble it.

  • The cylinder is bounded by two circles and the lateral intersection connecting them.
  • The two circles overlap and are parallel to each other (since they are three-dimensional, this means that the two planes in which the circles are also parallel to each other).
  • Each of the circles is called a base.
  • The distance between the bases is fixed and is called the height of the cylinder (in the drawings H H , it is actually also the width of the rectangle in the second drawing).
  • The radius of each of the two circles is defined as R R
  • Each cylinder is defined by its radius (R) (R) and height H H .

Calculation of Cylinder Volume

The volume of the cylinder is basically the volume trapped between the two bases and the side.

To calculate it we need the radius of the cylinder (R) (R) and the height of the cylinder (H) (H) .

In geometry it is customary to identify the volume with the letter (V) (V) .

The formula for the volume is:

V=π×R2×H V=\pi\times R²\times H

  • π (pi) is a mathematical constant whose value is (approximately) equal to. 3.14 3.14 .
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Calculation of Cylinder Area

When we talk about cylinder area, we must distinguish between total surface area and lateral surface area.

Total surface area is the sum of the areas of the two bases and the lateral area (identified by AA). The area of the base is πR2 \pi R²

And to obtain the area of the two bases we multiply by 2 2 , that is,

2XπR2 2X\pi R²

To calculate the lateral area, we return to the sliced form. The width of the rectangle is HH while the length of the rectangle (we identify it by LL ) is equal to the circumference of the circle. The circumference of the circle is calculated by the formula. R×L=2XR\times L=2X.

From here we get that the side area is the area of the sliced rectangle, and we must multiply the length of the rectangle by the width of the rectangle.

The area of the resulting rectangle is R×H=2XR\times H=2X.

To obtain the total surface area, we add the area of the two bases and the side area. We will eliminate a common factor outside the parentheses. R2XR 2X We obtain the following formula:

A=2XR(R+H)A=2XR(R+H)

Lateral surface area is the lateral surface area only, without the bases (identified as S). That is, we refer to the area of the sliced rectangle, which we have already calculated for the total surface area.

The formula is:

S=2XRHS= 2XRH


Cylinder Examples and Practice

Exercise 1

Given the cylinder shown in the drawing.

Exercise 1 Given the cylinder shown in the drawing

According to the data, find the volume of the cylinder, the lateral surface area and the total surface area.

Solution:

From the figure it can be seen that the radius of the bases is equal to R=5 R=5 cm, and the height of the cylinder is equal to H=10 H=10 cm.

Now it remains only to replace in the formulas that we have learned.

Calculation of the volume of the cylinder:
V=π×R2×H V=\pi\times R²\times H


=3.14×25×10=785= 3.14\times 25\times 10= 785

Lateral surface area:

S=2×π×R×H=2×3.14×5×10=314S= 2\times\pi\times R\times H = 2\times 3.14\times 5\times 10= 314

Total surface area:

A=2×π×R×(R+H)=2×3.14×5(5+10)=471A= 2\times\pi\times R\times (R+H) = 2\times 3.14\times 5( 5+10)= 471

Answer: Volume of the base 785cm3 785\operatorname{cm}^3 , lateral surface area 314 314 cm², total surface area 471 471 cm².


Exercise 2

Given the cylinder shown in the drawing.

We will use the formula for the volume of the cylinder

The area of the base is equal to 12 12 cm². According to the data, find the volume of the cylinder.

Solution:

From the figure it can be seen that the height of the cylinder is H=7H= 7 cm.

We will use the formula for the volume of the cylinder:

V=π×R2×H V=\pi\times R²\times H

Note that the base area is calculated by the formula. πR2 \pi R²

And in fact it is part of the volume formula.

The area of the base we know from the datum, it is equal to 12 12 cm², so we are left to place and obtain:

πR2=V=π×R2×H \pi R²=V=\pi\times R²\times H
XH=12×7=84X H = 12\times 7= 84

Answer: The volume of the cylinder is 84 84 cc.


If this article interests you, you may be interested in the following articles:

Cylinder volume

Right triangular prism

Bases of a right triangular prism

The lateral faces of a prism

Lateral edges of a prism

Height of a prism

Surface area of triangular prisms

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