In practice, tasks often arise that require the ability to construct sections of geometric shapes of various shapes and to find section areas. In this article, we will consider how important sections of a prism, pyramid, cone, and cylinder are constructed, and how to calculate their area.
Volumetric figures
From stereometry it is known that a three-dimensional figure of absolutely any type is limited by a number of surfaces. For example, for polyhedrons such as a prism and a pyramid, these surfaces are polygonal sides. For the cylinder and cone, we are already talking about the surfaces of rotation of the cylindrical and conical figures.
If we take a plane and intersect it arbitrarily in the surface of a three-dimensional figure, then we get a section. Its area is equal to the area of the part of the plane that will be inside the volume of the figure. The minimum value of this area is zero, which is realized when the plane touches the figure. For example, a section that is formed by a single point is obtained if the plane passes through the top of the pyramid or cone. The maximum value of the cross-sectional area depends on the relative position of the figure and the plane, as well as on the shape and size of the figure.
Below we consider how to calculate the area of the formed cross sections for two rotation figures (cylinder and cone) and two polyhedra (pyramid and prism).
Cylinder
A circular cylinder is a figure of rotation of a rectangle around any of its sides. The cylinder is characterized by two linear parameters: the radius of the base r and height h. The diagram below shows what a circular straight cylinder looks like.
There are three important section types for this shape:
- round;
- rectangular;
- elliptical.
The elliptical is formed as a result of the plane crossing the lateral surface of the figure at a certain angle to its base. Round is the result of the intersection of the secant plane of the side surface parallel to the base of the cylinder. Finally, a rectangular one is obtained if the secant plane is parallel to the axis of the cylinder.
The circular cross-sectional area is calculated by the formula:
S 1 = pi * r 2
The area of the axial section, that is, rectangular, which passes through the axis of the cylinder, is determined as follows:
S 2 = 2 * r * h
Cone sections
The cone is the figure of rotation of a right triangle around one of the legs. The cone has one top and a round base. Its parameters are also the radius r and height h. An example of a cone made of paper is shown below.
There are several types of conical sections. We list them:
- round;
- elliptical;
- parabolic;
- hyperbolic
- triangular.
They replace each other if you increase the angle of inclination of the secant plane relative to the round base. The easiest way to write the formula is the cross-sectional area of the round and triangular.
A circular section is formed as a result of the intersection of the conical surface with a plane that is parallel to the base. The following formula is valid for its area:
S 1 = pi * r 2 * z 2 / h 2
Here z is the distance from the top of the figure to the formed section. It can be seen that if z = 0, then the plane passes only through the vertex, therefore, the area S 1 will be equal to zero. Since z <h, the area of the studied section will always be less than its value for the base.
A triangle is obtained when a plane intersects a figure along its axis of rotation. The shape of the resulting section will be an isosceles triangle, the sides of which are the diameter of the base and the two generatrices of the cone. How to find the cross-sectional area of a triangular? The answer to this question will be the following formula:
S 2 = r * h
This equality is obtained if we apply the formula for the area of an arbitrary triangle through the length of its base and height.
Prism sections
Prism is a large class of figures that are characterized by the presence of two identical polygonal bases parallel to each other, connected by parallelograms. Any section of a prism is a polygon. In view of the diversity of the figures under consideration (inclined, straight, n-angular, regular, concave prisms), the diversity of their cross sections is great. Further, we consider only some special cases.
If the secant plane is parallel to the base, then the sectional area of the prism will be equal to the area of this base.
If the plane passes through the geometric centers of the two bases, that is, it is parallel to the side edges of the figure, then a parallelogram is formed in the section. In the case of direct and regular prisms, the considered sectional view will be a rectangle.
Pyramid
A pyramid is another polyhedron that consists of an n-gon and n triangles. An example of a triangular pyramid is shown below.
If the section is drawn parallel to the n-angular base by a plane, then its shape will be exactly equal to the shape of the base. The area of such a section is calculated by the formula:
S 1 = S o * (hz) 2 / h 2
Where z is the distance from the base to the section plane, S o is the area of the base.
If the secant plane contains the top of the pyramid and intersects its base, then we get a triangular section. To calculate its area, you need to use the appropriate formula for a triangle.