After reading this article, you will learn how to find the height of a 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 discusses all the necessary basic concepts and properties that are sure to come in handy in practice.
Fundamental theory
Before you can 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, though not necessarily, circular) to a point called a apex.
A cone is formed by a set of segments, rays or straight 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.
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 problems.
If the base is a polygon, then the object is a pyramid.
The segment connecting the vertex and the line,the bounding base is called the generatrix.
How to find the height of a cone
Let's approach the issue from the other side. Let's start with the volume of the cone. To find it, you need to calculate the product of the height with the third part of the area.
V=1/3 × S × h.
Obviously, from this you can get the formula for the height of the cone. It is enough just to make the correct algebraic transformations. Divide both sides of the equation by S and multiply by three. Get:
h=3 × V × 1/S.
Now you know how to find the height of a cone. However, you may need other knowledge to solve problems.
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 just a cone whose apex is at infinity.
The following properties only work for a right circular cone.
- Given the radius of the base r and the height h, then the formula for the area will look like this: P × r2. The final equation will change accordingly. V=1/3 × P × r2 × 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 conic sections.
There are often problems where it is necessary to use the formula for the volume of a truncated cone. It is derived from the usuallooks like this:
V=1/3 × P × h × (R2 + Rr + r2), where: r is the radius of the lower base, R is the upper one.
All this will be enough to solve a variety of examples. Unless you may need knowledge that is not related to this topic, for example, the properties of angles, the Pythagorean theorem and more.
