FACULTY OF NATURAL SCIENCES
CONSTANTINE THE PHILOSOPHER UNIVERSITY NITRA
ACTA MATHEMATICA 12
FINDING PROPERTIES OF PRISMS AND PYRAMIDS
WHILE SOLVING STEREOMETRIC PROBLEMS
JOANNA MAJOR, ZBIGNIEW POWĄZKA
ABSTRACT. In this article authors give some examples of stereometric tasks in which
afterthought on gained resolve leads to formulate mathematical theorem. These theorems are
often unknown to students and they exceed teaching curriculum.
1. Introduction
An important part of education in secondary schools is teaching the elements of flat
and spatial geometry with usage trigonometric functions. It creates possibility to practice
computational techniques and develope spatial imagination. There are known various didactical works relevant to this topic. (vide e.g. [1]).
Taking a look at stereometric task we mostly find tasks of designating area and volume
of the solid, often with usage values of angles between diagonals, edges or walls of the
solid. The solution of these tasks is often adequate formula. However, it turns out that
discussion of conditions which make the task feasible leads to the peculiar conclusions
which describes interesting properties of these solids.
In this article authors give some examples of stereometric tasks in which afterthought
on gained resolve leads to formulate mathematical theorem. These theorems are often
unknown to students and they exceed teaching curriculum.
We deem that solving mentioned type of tasks is very important in mathematical education because they create opportunity to develop mathematical creativity. The discussion
on gained solutions pursues Polya’s recommendation to “look back” (see [7]).
To start with, it is worth explaining the legitimacy of our using the word “find” in the
title of the paper. In the [8] we can read the following definitions of this term: “find” 1. to
trace or track down somebody or something hidden, 2. to discover somebody or something
by searching, 3. to discover by chance, 4. to state that something exists.
Z. Krygowska mentions four situations in which solving an appropriately selected
problem leads to the formulation of a theorem. In the first situation, the student solves a
problem presented to him or her a priori. The second case involves the student’s solving a
more open-ended problem; one that is clearly geared towards formulating and then proving
a new theorem. In the third situation, it is the student who asks questions and tries to answer them, either by himself or with the help of the teacher. Finally, it may well be that the
student himself formulates a theorem in the form a hypothesis, based either on his observations of some regular properties in many cases, or on his intuitive conviction that such
connections exist. The question that the student asks at this point, is about the general “accuracy” of “what he has noticed” (see [2]).
The term “to discover” a theorem is commonly used to denote a practice opposite to
simply presenting the student with a ready-made one (compare [5]). Throughout our paper,
we will use this term to mean “finding” the properties of a mathematical object, as a result
JOANNA MAJOR, ZBIGNIEW POWĄZKA
of a mathematical activity geared towards solving the problem presented, followed by reflecting on the solution.
Tasks which are presented below do not allow on full realisation of situations pointed
by Krygowska. There are formulated theorems in proposed tasks, which are relevant to
some mathematical objects, as the result of reflection on gained solutions. The essential
fact is that mathematical commands contained in tasks are relevant to different object than
objects mentioned in theorems. For example, the task goes for designate volume of cuboid,
while formulated theorem is relevant to dependence between values of angles which are
built by diagonal and edges of cuboid. A student who works with proposed tasks can't formulate adequate theorem a priori (before the task is resolved). In presented situations
proving of the theorem succeed before formulating it.
This form of mathematical activity requires from student to make advanced usage of
many concepts which could be asociate with discussed topic, in this case with prism or
pyramid. The proposal of this sort of tasks is very important element of mathematical
education because – as it is shown by our researches – operative using of concepts gives a
lot of difficulties to students (see [3] and [4]).
2. The propositions of tasks which lead to discovering theorems
Example 1.
Calculate the volume of the cuboid if there is known that diagonal which length amounts d:
a) builds with edges of this cuboid angles which amount is α, β, γ, (cf fig. 1),
b) builds with faces of this cuboid angles which amount is α, β, γ, (cf fig. 2),
c) builds with the base plane the angle γ and diagonals of spatial faces out-
going from the same vertex build with the base plane angles α, β (cf fig. 3).
Resolving of part a) of example 1 leads to discover theorem 1 and conclusion 1.
Theorem 1.
Let a, b, c be lengths of cuboid's edges and let d be the length of its diagonal. We
designate as α, β, γ, amounts of angles which are built by this diagonal and, in sequence, edges which lengths amount a, b, c. Then
a
b
c


 d.
cos  cos  cos 
Figure 1.
FINDING PROPERTIES OF PRISMS AND PYRAMIMIDS…
As for the cuboid with edges a, b, c and diagonal d equation a 2  b 2  c 2  d 2 , is
satisfied, the consequence of theorem 1 is following conclusion.
Corollary 1.
If diagonal of the cuboid builds with edges of this cuboid angles which amounts are α, β, γ,
then
cos2   cos 2   cos 2   1.
Resolving of the part b) of example 1 leads to discover theorem 2 and conclusions 2 and 3.
Theorem 2.
Let a, b, c be length of cuboid's edges, d length of its diagonal and α, β, γ, amounts of angles which are built by diagonal and faces of cuboid. Then
a
b
c


 d.
sin  sin  sin 
Figure 2.
In this case there is true the theorem analogous to the conclusion 1.
Corollary 2.
If diagonal of cuboid builds with edges of this cuboid angles which amounts are α, β, γ,
then
sin 2   sin 2   sin 2   1.
If we assumpt that the cuboid is inscribed into sphere which radius amounts R, then, diagonal of this cuboid is the radius of the sphere. As consequence of theorem 2 we can remark
following fact.
Corollary 3.
If cuboid which edges amount a, b and c is inscribed into sphere with radius R, then
a
sin 

b
sin 

c
sin 
 2R
where α, β, γ are amounts of angles which are built by radius of this sphere (connecting
two opposite vertexes of the cuboid) and adequate edges of this cuboid.
Forthcoming formula is analogous to law of sines on plane.
Resolving of part c) of example 1 leads to discover theorem 3.
JOANNA MAJOR, ZBIGNIEW POWĄZKA
Theorem 3.
Let a, b, c be lengths of cuboid’s edges, let α, β be amounts of angles which are built by
diagonals of lateral faces and base’s plane and let γ be the amount of the angle created by
diagonal of the cuboid and diagonal of the base. Then
a tg   b tg   a 2  b2 tg   c.
Figure 3.
This leads us to notice following relation between angles from the theorem 3.
Corollary 4.
If in the cuboid α, β are amounts of angles which are built by diagonals of lateral faces and
base’s plane and γ is the amount of the angle created by diagonal of the cuboid and diagonal of the base, then
ctg 2   ctg 2   ctg 2  .
Example 2.
Calculate the volume of tetragonal right regular prism in which diagonal of the base
amounts d and it builds angle α with lateral face’s diagonal which outgoing from the same
vertex.
Figure 4.
FINDING PROPERTIES OF PRISMS AND PYRAMIMIDS…
We take designations as at fig. 4.
We have
d 2 d 1  2 cos 2  d 3 1  2 cos 2 
.
V  a h 


2
2 cos 
4 cos 
2
It follows that 1  2cos2   0, hence cos 2  0 . As the triangle ACH is acuteangled and isosceles simultaneously, thus 2  2  34 . Therefore 4    34 .
Now then we obtain theorem 4.
Theorem 4.
At every right regular tetragonal prism amount of angle between lateral face’s diagonal and
base’s diagonal which outgoing from the same vertex belongs to bracket  π4 , π2  .


Example 3.
At right regular tetragonal prism edge of the base amounts a. Calculate the volume of this
prism if there is known that amount of angle built by lateral edge and base’s edge outgoing
from the same vertex is α.
Figure 5.
We take designations as at fig. 5.
The volume of prism amounts
1
1
a
a3
V   a2  h   a2 
tg 2   1 
tg 2   1 .
3
3
2
6


The angle α is acute angle so we obtain that the task has solution if and only if    4 ,2  .
The consequesnce of the task solution is theorem 5.
Theorem 5.
At every right regular tetragonal prism amount of acute angle at base of lateral face
belongs to bracket  4 , 2  .


The problems which are presented above could be extended and they should be treated
as patterns of building several sets of tasks. The usage of this kind of examples requires
knowledge and skills relevant to properties of trigonometric functions and creates opportunity to expand knowledge about amounts of angles in solids. These students who work
with described tasks obtain the chance to self-reliant finding and formulating theorems.
JOANNA MAJOR, ZBIGNIEW POWĄZKA
Lastly, it is worth to mark that solving of the task is in fact searching of undisclosed theorem and its proof.
Bibliography
[1.] Kartasiński S., Nauczanie trygonometrii, Państwowe Zakłady Wydawnictw
Szkolnych, Warszawa 1960.
[2.] Krygowska Z., Zarys dydaktyki matematyki, cz. 3, WSiP, Warszawa 1977.
[3.] Major, J., Rola zadań i problemów w kształtowaniu pojęć matematycznych na
przykładzie bezwzględnej wartości liczby rzeczywistej, Roczniki PTM, seria
V, Dydaktyka Matematyki 29, 2006, 297 - 310.
[4.] Major, J., Powązka, Z., Pewne problemy dydaktyczne związane z pojęciem
wartości bezwzględnej, Annales Academiae Pedagogice Cracoviensis Studia
Ad Didacticam Mathematicae Pertinentia I, Kraków 2006, 163 - 185.
[5.] Nowak W. , Konwersatorium z dydaktyki matematyki, PWN, Warszawa 1989.
[6.] Nowosiłow S., Specjalny wykład trygonometrii, Państwowe Wydawnictwo
Naukowe, Warszawa 1956.
[7.] Polya G., Jak to rozwiązać, Wydawnictwo Naukowe PWN, Warszawa 1993.
[8.]
Mały Słownik Języka Polskiego, red. Skorupka, S., Auderska, H., Łempicka,
Z., PWN, Warszawa 1969.
Joanna Major & Zbigniew Powązka
Pedagogical University of Cracow
Institute of Mathematics
Podchorążych 2
PL - 30-084 Cracow
e-mail: [email protected]
e-mail: [email protected]