Finding the Volume of Cylinder
Before getting to the main formula that is used to calculate the volume of cylinder, let’s learn more information about this geometrical figure, its elements, properties, and sections.
The Notion of Cylinder
There are many objects around us that are physical models of the cylinder. For example, many machine parts have a cylindrical shape or represent a combination thereof, and majestic columns of churches and cathedrals are also made in the shape of cylinders, which emphasizes their beauty and harmony.
Euclid thought of a cylinder as a figure that can be obtained by rotating the rectangle. So, cylinder is a geometric body bounded by a cylindrical surface and two parallel planes that intersect it. The formula for finding the volume of cylinder has been known in the Ancient Greece, where it was formed.
A cylindrical surface is the surface, obtained in such a reciprocating movement of a straight line (generator) in the space that the point of the generator moves along a flat curve (directrix). The part of the surface of the cylinder that is bounded by a cylindrical surface called the cylinder surface. Another part bounded by parallel planes is the base of the cylinder. Thus, the border of the base will be of the same shape as the directrix.
In most cases, a cylinder is understood as a right circular cylinder, where the directrix – the circle and the base is perpendicular to the generator. Such a cylinder has an axis of symmetry.
Other cylinder types (by the slope of the generator) can be oblique or sloping (if the generator doesn’t touch the base at right angles). Due to the shape of the base, cylinders can be elliptical, hyperbolic, and parabolic.
Prism is also a kind of the cylinder with a base in the form of a polygon.
Another definition for a cylinder is that it is the body, which is formed by the rotation of a rectangle around the line, including its side.
Elements and Properties of the Cylinder
Circles are called bases of the cylinder, and the segments joining the corresponding points of the circles are the generatrixes of a cylinder.
Since parallel transportation is a movement, the bases of the cylinder are equal.
Since during the parallel transformation a plane moves parallel to the plane (or in itself), the bases of the cylinder lie in parallel planes.
Since during the parallel transformation the points are shifted by parallel (or matched) lines on the same distance, the generatrixes of the cylinder are parallel and equal.
The surface of a cylinder consists of bases and the side surface. The side surface is made up of generators.
The cylinder is direct if its generators are perpendicular to the plane of the bases.
Visually, a direct cylinder can be represented as a geometric body that describes a rectangle while rotating around its side.
The radius of the cylinder is the radius of its base. Cylinder’s altitude represents the distance between the planes of its bases. Cylinder’s axis represents a straight that goes through the center of the base. This line is parallel to the generating lines.
Cylinder is equilateral if its height is equal to the diameter of the base.
If the bases of the cylinder are plane (and therefore the planes containing them are parallel), then this is a cylinder standing on a plane. If the bases of the cylinder standing on a plane are perpendicular to the generating line, the cylinder is direct.
In particular, if the base of the cylinder standing on the plane is circle, then this cylinder is circular. If in the base of the cylinder is ellipse, then this is the elliptical cylinder.
Sections of the Cylinder
The section of the cylinder with a plan, which is parallel to the axis, is a rectangle. Two sides of this rectangle form a cylinder, while the other two sides are parallel chords to the bases.
In particular, the axial section is a rectangle. This is the section of the cylinder with the plane that passes through its axis.
The section of a cylinder with plane parallel to the ground is circle.
The section of a cylinder with a plane that is not parallel to the base and the axis is oval.
Finding the Volume of Cylinder
- The first step in finding the volume of cylinder is to find the base radius. As we know, both bases are equal. If the radius is given, go to the next step. Otherwise, you need to measure the circle in its widest part, to find its diameter. To get the radius you need to divide the diameter by 2 to. For example, the radius of the cylinder is 1 cm.
If you have a value of a diameter, divide it by 2 to get the radius.
If you have a circumference, divide it by 2 pi to get the radius.
- The next step in finding the volume of cylinder is to find the area of the base according to the formula: A = pi r2. Just substitute in her range. Here’s how: A = pi x 12 = A = pi x 1.Since pi ≈ 3,14, the base area is 3.14 cm2.
- The third step in calculating the volume of cylinder is to find the height of the cylinder. If it is given in you example, go to the next step. Otherwise measure it with a ruler. As we know, height of the cylinder is the distance between the two bases. For example, the height is 4 cm.
- Finally, the last step to calculate the volume of cylinder is to multiply the area of the base by the height of the cylinder to find the volume of cylinder. The base area is equal to 3.14 cm 2, and the height is 4 cm, so 3.14 cm2 x 4 cm = 12.56 cm 3 is the volume of cylinder. The volume is measured in cubic units, as this is the quantity that characterizes volumetric (three dimensional) shapes.
Advice for Finding the Volume of Cylinder
- Think of several examples for finding the volume of cylinder and practice. This will help you better learn the formula for the volume of cylinder.
- To find the right volume of cylinder, measure everything accurately.
- Use the calculator if it is difficult to calculate the volume of cylinder without it.
- Remember that the diameter is the greatest distance between two points on the circle. In order to find the diameter put a ruler with a starting point to the edge of the cylinder and find the greatest distance to the other end. This will be the diameter.
- In general cases, the volume of cylinder is always equal to the product of the base area by its height (however, this does not work in some cases, for example in the cone).
- To find the radius, it is much easier to find the diameter and divide it by two.
- The volume of cylinder is calculated according to the formula: V = pir2h.
- Multiplication of the base area by the height is similar to the addition of many of the cylinder’s bases (imagine a transparent cylinder filled with its bases).
Formulas for Calculating the Volume of Cylinder of Various Kinds
There are two formulas that can help you find the volume of cylinder that is inclined:
- Volume of cylinder is the length of the generatrix of the cylinder multiplied by the sectional area of a plane perpendicular to the generatrix: V = Sl.
- The volume of cylinder is equal to the area of the base times the height (distance between the planes in which the bases lie): V = Sh = Sl sin f, where l is the length of the generatrix, and f is the angle between the generator and the plane of the base.
For direct cylinder, h = l.
For direct cylinder, sin f = 1, l = h and S = S, so the volume of cylinder is equal to V = Sl = Sh.
For a circular cylinder, the formula for the volume of cylinder is V = pi R2h = pi x d2/4 x h, where d is the diameter of the base.
Application of Cylinders
Many objects that surround us are cylinders. Pots, jars for bulk products, lamps, vases, bottles, cans, cars, details, and crayons are all represent cylinders in one or another of their form. It is very important to know the formula of volume of cylinder in order to be able to build various details, buildings, and other objects of the right size. The volume of cylinder formula is considered to be one of the simplest in geometry.
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