Geometry Surface of a triangle Surface of an equilateral triangle Surface of an isosceles triangle Surface of a scalene triangle The hypotenuse of a right-angled triangle Side of a right-angled triangle Surface of a right-angled triangle Diagonal of a square Surface of a square Diagonal of a rectangle Surface of a rectangle Surface of a rhombus Surface of a parallelogram Surface of a trapezoid Surface of a regular pentagon Regular and irregular polygons Surface of a regular hexagon Perimeter of a circle Surface of a circle Length of an arc of circle Surface of a circular sector Surface of a circular ring length of a propeller Surface of a segment of parabola Surface of an ellipse volume of a cube Volume of a parallelepiped Volume of a cylinder Volume of a hollow roll Volume of a ring with square section Volume of a torus Surface of a sphere Volume of a sphere Surface of a segment of a sphere Volume of a segment of a sphere Volume of a paraboloid Return to the synopsis To contact the author Low of page

Created it, 05/10/15

Update it, 05/10/30

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MATHEMATICAL FORMS “GEOMETRY” 1st PART

FOOT- NOTE : In these lessons with for goal to represent and gather formulas, tables and graphs usable in elementary calculations of geometry, physics and electronics.

They were conceived as of the memories assistances of reference to which one will be able to refer to solve problems relating to the calculation of the circuits. It is thus not necessary to remember exactly the formulas and the procedures of computation of the exposed subjects. It will be enough to only once carry out the calculations indicated in the examples to remember to the convenient period of the existence of a formula or a graph usable for the solution of a given problem. Indeed, even if one remembers only very vaguely procedure, it will be always possible to resort to these lessons of memories assistances where the formulas and graphs are indicated which will make it possible to solve the practical problems encountered in the work of the technician.

However, if you wish to be exerted with calculations, you will be able to at will make new exercises by working out them on the model of the examples deferred in these lessons of memories assistances.

To this end, it will be enough to replace the numerical values of the example by other values chosen randomly and to carry out the operations with the new data; in the more complex cases, you will be able to check the exactitude of the final result with the evidence of arithmetic calculation.

To facilitate the reading of these lessons of memories assistances, the formulas are numbered in an order ascending, as well as the tables and graphs.

Each formula is illustrated a practical example of application.

In general, one proceeds in the following way: after having selected the formula, one replaces the letters of the second member by the respective numerical values (given) and one carries out calculations to obtain the final result. In the more complex cases, it will be indicated an additional control which will be able to be added to the evidence of arithmetic calculation.

Sometimes, it can prove to be advantageous to replace ordinary calculation, i.e. numerical, by a known particular procedure under the name of graphic method; therefore, in addition to the ordinary calculation which derives directly from the formula, one will be able to sometimes take into account also that carried out by the graphic method.

GEOMETRY

It can happen that one needs to know dimensions, surfaces or unspecified volumes of objects, and that when it is not easy, or even completely impossible, to take direct measurements. It is then necessary to carry out calculations.

For example, it can present cases where it is necessary to know the length of a whorl, the section of a driver, the section or the volume of a magnetic core…

In general, they are problems which one can quickly solve by applying a suitable formula of geometry.

We will thus find in this lesson of memory assistance the formulas of geometry having a practical application in electronics.

FORMULATE 1 - Calculation of the area of a triangle knowing the values of the base and height (figure 1-a).

FORMULATE 2 - Calculation of the area of an equilateral triangle, “triangle having three equal sides” (figure 1-b) knowing the length on the side.

Example (figure 1-b) :

Data : c = 5 cm

Surface : S 0,433 x 52 = 0,433 x 25 = 10,825 cm2

FORMULATE 3 - Calculation of the area of an isosceles triangle “triangle having two equal sides” knowing the value on the equal sides and the base.

FORMULATE 4 - Calculation of the area of a scalene triangle “triangle having three unequal sides” knowing the length on the sides.

In this formula “p” indicates the half-perimeter, i.e. the half the sum on the three sides. For applying the formula, it is necessary to calculate separately the value “p” of the half-perimeter.

FORMULATE 5 - Calculation of the hypotenuse of a right-angled triangle knowing the two other sides (the right-angled triangle is a triangle having an angle of 90°; the hypotenuse is the largest side, the two other sides form the angle of 90°). (See the figure 1-e above).

FORMULATE 6 - Calculation on a side of a right-angled triangle knowing the lengths of the hypotenuse and other side (for the significance of the terms, you defer to formula 5).

FORMULATE 7 - Calculation of the area of a right-angled triangle knowing the two sides of the right angle.

FORMULATE 8 - Calculation of the diagonal of a square knowing the length on the side. (Figure 2-a).

FORMULATE 9 - Calculation of the area of a square knowing the length on the side.

Example (figure 2-a) :

Data : c = 50 mm

Surface: S = 502 = 2 500 mm2

FORMULATE 10 - Calculation of the area of a square knowing the length of the diagonal.

S = d2 / 2

S = surface

d = diagonal

Example (figure 2-a) :

Data : d 70,70 mm (approximate value established with formula 8)

Surface: S 70,702 / 2 = 4 998,49 / 2 = 2 499,245 mm2

Compare this result with that obtained by applying formula 9. The difference in 0,755 mm2 (2 500 - 2 499,245 = 0,755) is due to the introduction of the approximate value of 70,70 into the calculation of the area, but the error which results from it is very weak (only of 0,03%), therefore practically negligible.

(To facilitate the reading, we defer the same figure below to knowing figure 2).

FORMULATE 11 - Calculation of the diagonal of a rectangle knowing the values of the base and height.

(This formula above is similar to formula 5).

FORMULATE 12 - Calculation of the area of a rectangle knowing the values of the base and height.

S = b x h

S = surface

b = bases

h = height

Example (figure 2-b) :

Data : b = 10 cm ; h = 5 cm

Surface: S = 10 x 5 = 50 cm2

FORMULATE 13 - Calculation of the area of a rhombus knowing the length of the diagonals (the rhombus is a quadrilateral having four equal sides and of the unequal adjacent angles).

S = D x d / 2

S = surface

D = large diagonal

d = small diagonal

Example (figure 2-c) :

Data : D = 8 cm ; d = 5 cm

Surface: S = 8 x 5 / 2 = 40 / 2 = 20 cm2

FORMULATE 14 - Calculation of the area of a parallelogram knowing the values of the base and height.

S = b x h

S = surface

b = bases

h = height

(This formula above is similar to formula 12).

Example (figure 2-d) :

Data : b = 15 cm ; h = 6 cm

Surface: S = 15 x 6 = 90 cm2

FORMULATE 15 - Calculation of the area of a trapezoid knowing the values of the two bases and height.

FORMULATE 16 - Calculation of the area of a regular pentagon knowing the length on the sides (the regular pentagon is a polygon having five equal sides and five equal angles).

S1,72 c2

S = surface

c = side

Example (figure 3-a) :

Data : c = 20 mm

Surface: S 1,72 x 202 = 1,72 x 400 = 688 mm2

FORMULATE 16 - 1 : Regular and irregular polygons
It is said that a polygon is regular when all its sides and all its angles are adequate (equal).

It is said that a polygon is irregular when certain on its sides and some of its angles are unequal (incongruous).

Regular polygon                   Irregular polygon

FORMULATE 17 - Calculation of the area of a regular hexagon knowing the length on a side (the regular hexagon is a polygon having six equal sides and six equal interior angles).

S = 2,60 x c2

S = surface

c = side

Example (figure 3-b “above”)

Data : c = 12 mm

Surface : S 2,60 x 122 = 2,60 x 144 = 374,4 mm2

FORMULATE 18 - Calculation of the perimeter of a circle (circumference) knowing the value of the diameter.

FORMULATE 19 - Calculation of the area of a circle knowing the value of the diameter.

FORMULATE 20 - Calculation length of an arc of circle knowing the value of the angle in the center and the length of.

(To facilitate the reading, we defer the same figure to knowing figure 3)

FORMULATE 21 - Calculation of the area of a circular sector knowing the value of the angle in the center and the length of the ray (a circular sector is the plane surface delimited by an arc of circle and two rays).

FORMULATE 22 - Calculation of the area of a circular ring knowing the value of the two diameters (a circular ring is the plane surface ranging between two concentric circumferences).

FORMULATE 23 - Calculation of the area of a segment of parabola knowing the value of the base and height (one calls segment of parabola the plane surface ranging between an arc of parabola and the cord underlain between the ends of the arc).

S = 2 / 3 x b x h

S = surface

b = bases

h = height
Example (figure 4-a) :
Data : b = 12 cm ; h = 8 cm

Surface : S = 2 / 3 x 12 x 8 = 2 / 3 x 96 = (2 x 96) / 3 = 64 cm2

FORMULATE 24 - Calculation of the area of an ellipse knowing the length of the two axes.

FORMULATE 25 - Calculation length of a propeller knowing the number of whorls, the values of the diameter and height.

FORMULATE 26 - Calculation of the volume of a cube knowing the length of the edge.

V = a3

V = volume

a = edge

Example (figure 5-a) :

Data : = 4 cm

Volume : V = 43 = 4 x 4 x 4 = 64 cm3

FORMULATE 27 - Calculation of the volume of a parallelepiped knowing the values length and width of the base, and the height.

V = a x b x h

V = volume

a = length of the base has

b = width of the base

h = height
Example (figure 5-b) :
Data : a = 25 mm ; b = 30 mm ; h = 70 mm

Volume : V = 25 x 30 x 70 = 52 500 mm3 = 52,5 cm3

FORMULATE 28 - Calculation of the volume of a cylinder knowing the values of the diameter and height.

FORMULATE 28 - 1: To calculate a cylinder of a volume generated by the rotation of a rectangle around one on its sides (side surface = 2Rh ; total surface = 2R (h + R) ; volume = R²h, h being the height and R the radius of the basic circle).

FORMULATE 29 - Calculation of the volume of a hollow roll knowing the values of the two diameters and height.

FORMULATE 30 - Calculation of the volume of a ring with square section knowing the values of the external diameters and interns.

FORMULATE 31 - Calculation of the volume of a torus (ring with circular section) knowing the value of the diameter external and that of the diameter of the section of the ring.

FORMULATE 32 - Calculation of the area of a sphere knowing the value of the diameter.

Example (figure 7-a) :
Data : d = 15 mm

Surface: S 3,14 x 152 = 3,14 x 225 = 706,5 mm2

FORMULATE 33 - Calculation of the volume of a sphere knowing the value of the diameter.

Example (figure 7-a) :
Data : d = 15 mm

Volume : V 0,523 x 153 = 0,523 x 3375 = 1765,125 mm3

FORMULATE 34 - Calculation of the area of a segment of a sphere knowing the values of the diameter of contour and height.

FORMULATE 35 - Calculation of the volume of a segment of a sphere knowing the value of the diameter of the base and height.

FORMULATE 36 - Calculation of the volume of a paraboloid knowing the value of the diameter of the base and height.

Daniel