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Translations of Encyclopedia about Mathematics


Volume and Surface Areas of Polyhedrons

Objects are three-dimensional shapes, such as cubes or spheres.

The basis of an object is its bottom surface.

The upper base is the surface area which lies opposite the lower base, which is the basis. If these two surfaces are parallel to one another, we refer to the distance between them as height.

All bordering surfaces of an object makes up that object’s surface. The shell (the lateral surface of an object) is all surface areas of an object between the lower and upper bases.

As with polygons, whose circumference is given by individually bordering abscissae, the surface of an object is calculated by the sum of the individual surface areas engulfing the object.

These individual surface areas meet at the edges of the object. Where three or more surfaces come together, we find a corner, or vertex. The position of the edges amongst themselves depends on the size of the angles which contain the individual surface areas.

The area which is contained within an object makes up that object’s volume.

The connecting lines between two vertexes which are not located on the surface of an object are called internal diagonals.

Objects are usually limited by known surface shapes, such as a cube, which is bordered by six sides, each made up of the same-sized square. On the other hand, a spire has a square shaped basis and triangular lateral sides. However, spheres do not have any flat surface, which is why we say it has a curved surface.

Cones and cylinders also have curved surfaces. While spheres do not have any flat surface, a cone has one round surface and a cylinder has two round bases.

Objects with flat, bordering surfaces are referred to as polyhedrons, which include such objects as platonic objects, which are objects like tetrahedrons, contained within same-sized surfaces.

When playing certain sports, one can sometimes find different shaped cubes, as shown in the picture examples.

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Volume and Surface Areas of Round Objects

Spheres, cones and cylinders are round objects which have curved surfaces - essentially imaginary polyhedrons with an unlimited number of surface areas. Based on this conception, equations exist to calculate such objects’ surface areas.


A sphere is bordered by one curved surface. All points of this surface have the same distance from the sphere’s centre (S) and have a distance of radius r.

The surface area of a sphere is: P = 4 . p . r2
The volume of a sphere is V = (4 . p . r

Spherical segments cut a section of the sphere to give us a flat surface. The spherical top of such a spherical segment is called the spherical cap.

If however the slice of the sphere has a conical shape, meaning that it has circular circumference and tapers at the centre, we have a sector of sphere.

If a sphere is dissected by two parallel running surfaces to give us a layer, we call it a spherical layer. The surface of such a spherical layer is called a spherical belt.

If a sphere is dissected by two, non-parallel running surfaces passing through each other in the centre of the sphere, we call the result a spherical wedge and the outer border of the spherical surface a spherical double-square.


A circular cone is an object with a circular basis and round shell which converges with a vertex with no upper base. With straight cones, the vertex S lies directly above the centre of the base circle. This is not the case with slanted cones.

The volume of a cone is one third the product of its height and the surface are of its basis:

V = 1/3 p r2 . h

Frustums have sliced-off vertexes.

If the lower and upper bases of some object are the same size, we call it a cylinder. The distance between both bases is referred to as height h. If the bases are lined up directly over one another, it is a straight cylinder, otherwise a slanted cylinder. The surface area between both bases is called the shell. A prism whose surface is made of an unlimited number of edges approaches the shape of a cylinder.

The volume V of a straight cylinder is: V = p . r2 . h
The surface area P of a straight cylinder is P = 2 p .r.h

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Platonic Objects

There are five types of platonic objects. Platonic objects are normal objects, such as normal polyhedrons (equilateral objects).

Tetrahedron: actually a pyramid with a triangular basis, where all triangles are equilaterals.
a regular six-sided object whose walls are squares of the same size. A cube has 12 equally long edges and eight right angle corners.
a polyhedron whose sides are made of eight equilateral triangles of the same size.
an object made of 12, equally large, regular pentagons.
a polyhedron made of 20 equilateral, same-sized triangles.

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