- Parts of the ortohedron
- Orthohedron formulas
- Area
- Volume
- Internal diagonal
- Examples
- - Example 1
- Solution to
- Solution b
- Solution c
- Solution d
- - Exercise 2
- Solution
- References
The orthohedron is a volumetric or three-dimensional geometric figure that is characterized by having six rectangular faces, so that the opposite faces are in parallel planes and are identical or congruent rectangles. On the other hand, the faces adjacent to a given face are in planes perpendicular to that of the initial face.
The orthohedron can also be considered as an orthogonal prism with a rectangular base, in which the dihedral angles formed by the planes of two faces adjacent to a common edge measure 90º. The dihedral angle between two faces is measured on the intersection of the faces with a perpendicular plane common to them.
Figure 1. Orthohedron. Source: F. Zapata with Geogebra.
Likewise, the ortohedron is a rectangle parallelepiped, since this is how the parallelepiped is defined as the volumetric figure of six faces, which are parallel two by two.
In any parallelepiped the faces are parallelograms, but in the rectangular parallelepiped the faces must be rectangular.
Parts of the ortohedron
The parts of a polyhedron, like the orthohedron, are:
-Aristas
-Vertices
-Faces
The angle between two edges of one face of the orthohedron coincides with the dihedral angle formed by its other two faces adjacent to each of the edges, forming a right angle. The following image clarifies each concept:
Figure 2. Parts of an ortohedron. Source: F. Zapata with Geogebra.
-In total an ortohedron has 6 faces, 12 edges and 8 vertices.
-The angle between any two edges is a right angle.
-The dihedral angle between any two faces is also right.
-In each face there are four vertices and at each vertex there are three mutually orthogonal faces.
Orthohedron formulas
Area
The surface or area of an ortohedron is the sum of the areas of its faces.
If the three edges that meet at a vertex have measures a, b, and c, as shown in Figure 3, then the front face has area c⋅b and the bottom face also has area c⋅b.
Then the two lateral faces have area a⋅b each. And finally, the floor and ceiling faces have area a tienenc each.
Figure 3. Orthohedron of dimensions a, b, c. Internal diagonal D and external diagonal d.
Adding the area of all the faces gives:
Taking a common factor and ordering the terms:
Volume
If the ortohedron is thought of as a prism, then its volume is calculated like this:
In this case, the floor of dimensions c and a is taken as the rectangular base, so the area of the base is c⋅a.
The height is given by the length b of the edges orthogonal to the faces of sides a and c.
Multiplying the area of the base (a⋅c) by the height b gives the volume V of the ortohedron:
Internal diagonal
In an orthohedron there are two kinds of diagonals: the outer diagonals and the inner diagonals.
The external diagonals are on the rectangular faces, while the internal diagonals are the segments that join two opposite vertices, being understood by opposite vertices those that do not share any edge.
In an orthohedron there are four internal diagonals, all of equal measure. The length of the internal diagonals can be obtained by applying the Pythagorean theorem for right triangles.
The length d of the external diagonal of the floor face of the ortohedron fulfills the Pythagorean relation:
d 2 = a 2 + c 2
Similarly, the interior diagonal of measure D fulfills the Pythagorean relationship:
D 2 = d 2 + b 2.
Combining the two previous expressions we have:
D 2 = a 2 + c 2 + b 2.
Finally, the length of any of the internal diagonals of the orthohedron is given by the following formula:
D = √ (a 2 + b 2 + c 2).
Examples
- Example 1
A bricklayer builds a tank in the shape of an ortohedron whose internal dimensions are: 6 mx 4 m in base and 2 m in height. It asks:
a) Determine the interior surface of the tank if it is completely open at the top.
b) Calculate the volume of the interior space of the tank.
c) Find the length of an interior diagonal.
d) What is the capacity of the tank in liters?
Solution to
We will take the dimensions of the rectangular base a = 4 m and c = 6 m and the height as b = 2 m
The area of an ortohedron with the given dimensions is given by the following relationship:
A = 2⋅ (a⋅b + b⋅c + c⋅a) = 2⋅ (4 m⋅2 m + 2 m⋅6 m + 6 m⋅4 m)
That is to say:
A = 2⋅ (8 m 2 + 12 m 2 + 24 m 2) = 2⋅ (44 m 2) = 88 m 2
The previous result is the area of the closed orthohedron with the given dimensions, but since it is a tank completely uncovered in its upper part, to obtain the surface of the inner walls of the tank, the area of the missing lid must be subtracted, which is:
c⋅a = 6 m ⋅ 4 m = 24 m 2.
Finally, the interior surface of the tank will be: S = 88 m 2 - 24 m 2 = 64 m 2.
Solution b
The interior volume of the tank is given by the volume of an orthohedron of the interior dimensions of the tank:
V = a⋅b⋅c = 4 m ⋅ 2 m ⋅ 6 m = 48 m 3.
Solution c
The interior diagonal of an octahedron with the dimensions of the interior of the tank has a length D given by:
√ (a 2 + b 2 + c 2) = √ ((4 m) 2 + (2 m) 2 + (6 m) 2)
Carrying out the indicated operations we have:
D = √ (16 m 2 + 4 m 2 + 36 m 2) = √ (56 m 2) = 2√ (14) m = 7.48 m.
Solution d
To calculate the capacity of the tank in liters, it is necessary to know that the volume of a cubic decimeter is equal to the capacity of a liter. It had previously been calculated in volume in cubic meters, but it has to be transformed to cubic decimeters and then to liters:
V = 48 m 3 = 48 (10 dm) 3 = 4,800 dm 3 = 4,800 L
- Exercise 2
A glass aquarium has a cubic shape with a side 25 cm. Determine the area in m 2, the volume in liters, and the length of an interior diagonal in cm.
Figure 4. Cubic-shaped glass aquarium.
Solution
The area is calculated using the same orthohedron formula, but taking into account that all the dimensions are identical:
A = 2⋅ (3 a⋅a) = 6⋅ a 2 = 6⋅ (25 cm) 2 = 1,250 cm 2
The volume of the cube is given by:
V = a 3 = (25 cm) 3 = 15.625 cm 3 = 15.625 (0.1 dm) 3 = 15.625 dm 3 = 15.625 L.
The length D of the inside diagonal is:
D = √ (3a 2) = 25√ (3) cm = 43.30 cm.
References
- Arias J. GeoGebra: Prisma. Recovered from: youtube.com.
- Calculation.cc. Exercises and solved problems of areas and volumes. Recovered from: calculo.cc.
- Salvador R. Pyramid + orthohedron with GEOGEBRA (IHM). Recovered from: youtube.com
- Weisstein, Eric. "Orthohedron". MathWorld. Wolfram Research.
- Wikipedia. Orthohedron Recovered from: es.wikipedia.com