After reading this article, you will learn how to find the height of the cone. The material presented in it will help to better understand the issue, and the formulas will be very useful in solving problems. The text contains all the necessary basic concepts and properties that will be useful in practice.
Fundamental theory
Before you find the height of the cone, you need to understand the theory.
A cone is a shape that tapers smoothly from a flat base (often, although not necessarily circular) to a point called a vertex.
A cone is formed by a set of segments, rays or lines connecting a common point with the base. The latter can be limited not only to a circle, but also to an ellipse, parabola or hyperbola.
An axis is a straight line (if any) around which the figure has circular symmetry. If the angle between the axis and the base is ninety degrees, then the cone is called straight. It is this variation that is most often found in tasks.
If the basis is a polygon, then the object is a pyramid.
The line connecting the top and the line bounding the base is called the generatrix.
How to find the height of the cone
We will approach the issue from the other side. First, use the volume of the cone. To find it, you need to calculate the product of height with the third part of the area.
V = 1/3 × S × h.
Obviously, from this one can obtain the formula for the height of the cone. It is enough to make the correct algebraic transformations. Divide both sides of the equality by S and multiply by three. We get:
h = 3 × V × 1 / S.
Now you know how to find the height of the cone. However, to solve problems you may need other knowledge.
Important formulas and properties
The material below will definitely help you in solving specific problems.
The center of mass of the body is on the fourth part of the axis, starting from the base.
In projective geometry, a cylinder is simply a cone whose vertex is at infinity.
The following properties only work for a straight circular cone.
- Given the radius of the base r and height h, then the formula for the area will look like this: P × r 2 . The final equation will change accordingly. V = 1/3 × P × r 2 × h.
- You can calculate the lateral surface area by multiplying the number "pi", the radius and the length of the generatrix. S = P × r × l.
- The intersection of an arbitrary plane with a figure is one of the conical sections.
Often there are problems where it is necessary to use the formula for the volume of a truncated cone. It is derived from the usual and has the following form:
V = 1/3 × P × h × (R 2 + Rr + r 2 ), where: r is the radius of the lower base, R - upper.
All this will be enough to solve a variety of examples. Unless knowledge may be needed that is not related to this topic, for example, the properties of angles, the Pythagorean theorem and more.