Before proceeding with the calculation of any values, it is necessary to determine the terminology. What is the cylinder? This geometric body consists of a lateral cylindrical surface and 2 base areas. The first is formed by a plurality of segments parallel to each other - forming a cylinder. They connect the upper and lower circles - the base of the shape. In this case, we are talking about a circular cylinder, which can be both inclined and straight. In addition, there are other cylindrical surfaces of 2 orders of order - elliptical, hyperbolic and parabolic cylinders.
Cylinder - Mathematical Calculation of Volume
In the case when the forming bulk figures are equal and perpendicular to the bases, we have a direct circular cylinder. If the angle between forming and planes, in which the circles are located, differ from 90 ° - the cylinder is inclined.
Circular cylinder direct
This body is a bulk geometric shape obtained by rotating the rectangle around one of the sides. The latter acts as an axis. We define the volume of this geometric body.
V \u003d S * H, where
- V - final desired value (volume),
- S - the foundation area of \u200b\u200bthe figure,
- H is the height of the figure.
In the case of a circular cylinder, the grounds are circles, from which it follows that it is necessary to calculate the area of \u200b\u200bthe circle. To do this, we use the formula S \u003d πr 2, where π is a constant value, the value of which is ≈3.14, and R is the radius of the circle lying at the base of the shape. Based on the ratio R \u003d D / 2, where D is the diameter of the base, the base area can be calculated by the formula S \u003d πd 2/4.
In view of the above, the determination of the volume of the cylinder requires the determination of its parameters such as the height and radius (or diameter) of the base. If the cylinder is straight, then the length of its guide is equal to its height.
Inclined circular cylinder
For this bulk figure, the ratio is still fair for finding the volume, with the only difference that its height is not equal to the guide. How to determine the height of the shape if the value of the generator is at the disposal? From the center of the upper base, omit perpendicular (it is height) to the lower plane (in which the second base of the cylinder is located). As a result, a rectangular triangle was formed (its parties - the height of the cylinder, the radius of its base, guiding). We use the definition of one of the trigonometric functions. sin\u2061α \u003d h / l, i.e. h \u003d sin\u2061α * l.
Here:
- h - desired height,
- l is the length of the guide,
- α is the angle between the guide and the foundation plane.
This kind of calculation is relevant when solving mathematical tasks based on available data.
Cylinder - Practical measurements of volume
The following technique will help if you have a task to determine the volume of the cylinder, using a submitted measuring means - a ruler. With it, you can easily determine the height of the geometric body, but what about the radius of the circle?
The rectangle can be entered into any circumference, moreover, its hypotenuse will be the diameter of the initial circle. We use this ratio. Take a rectangular sheet of paper and place a vertex A of its angle (90 °) on the circumference arc. Mark on the point of intersection point of the parties of the rectangle (leaving the vertex a) with a circle. Connect these points on a sheet of paper. The length of the resulting segment and will be equal to the diameter of the base circle. Returning to the formula V \u003d S * H \u200b\u200b\u003d Hπd 2/ 4, find out the desired value.
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