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**Geometry Form : **

** NOTE :** In these lessons, we aim to represent and collect formulas, tables and graphs that can be used in elementary calculations of geometry, physics and electronics.

They have been designed as reference reference aids that can be used to solve problems relating to the calculation of circuits. It is therefore not necessary to remember exactly the formulas and calculation procedures of the exposed subjects. It will be sufficient to carry out once the calculations indicated in the examples to remember at the opportune moment of the existence of a formula or a chart usable for the solution of a certain problem. Indeed, even if one remembers very vaguely a procedure, it will always be possible to use these lessons of aide memoirs which are indicated formulas and charts that will solve the practical problems encountered in the work technician.

However, if you want to practice calculations, you can do new exercises at will by developing them on the model of the examples reported in these lessons aids.

For this purpose, it will suffice to replace the numerical values of the example with other values chosen at random and to carry out the operations with the new data **;** in more complex cases, you can check the accuracy of the final result with the proofs of arithmetic calculation.

**To make it easier to read these lesson aids, the formulas are numbered in ascending order, as well as tables and graphs.**

Each formula is illustrated with an example of practical application.

In general, one proceeds as follows **:** after having selected the formula, the letters of the second member are replaced by the respective numerical values (data) and the calculations are carried out to obtain the final result. In more complex cases, it will be indicated an additional control that may be added to the proofs of the arithmetic calculation.

Sometimes, it may be advantageous to replace the ordinary, that is to say numerical, calculation by a special procedure known as the graphical method **;** hence, besides the ordinary calculus which derives directly from the formula, one can sometimes take into consideration also that carried out by the graphical method.

It may happen that one needs to know the dimensions, surfaces or volumes of any objects, and that when it is not easy, or even quite impossible, to make direct measurements. We must then proceed to calculations.

For example, there may be cases where it is necessary to know the length of a turn, the section of a conductor, the section or the volume of a magnetic core ...

In general, these are problems that can be quickly solved by applying an appropriate formula of geometry.

In this lesson, we will find geometry formulas with a practical application in electronics.

**
FORMULA 1
- **Calculation of the surface of a triangle knowing the values of the base and the height (Figure 1-a).

**Calculate the area of a triangle (Figure 1-a)**

**
FORMULA 2
- **Calculation of the area of an equilateral triangle, "triangle having three equal sides" (Figure 1-b) knowing the length of the side.

**Calculate the area of an equilateral triangle (Figure 1-b)**

**Example (Figure 1-b) :**

**Data : c =** **5 cm**

**Area : S
0,433 x 5 ^{2} =**

**
FORMULA 3
- ** Calculation of the area of an isosceles triangle "triangle having two equal sides" knowing the value of the equal sides and the base.

**Calculate the area of an isosceles triangle (Figure 1-c)**

**
FORMULA 4
- **Calculation of the surface of a scalene triangle "triangle having three unequal sides" knowing the length of the sides.

**Calculate the area of a Scalene triangle (Figure 1-d)**

In this formula **"p"** designates the half-perimeter, that is to say the half-sum of the three sides. Before applying the formula, the value **"p"** of the half-perimeter must be calculated separately.

**
FORMULA 5
- **Calculation of the hypotenuse of a right triangle knowing the other two sides (the right triangle is a triangle with an angle of 90°, the hypotenuse is the largest side, the other two sides are the angle of 90°). (See Figure 1-e above).

**Calculate the hypotenuse of a right triangle (Figure 1-e)**

**
FORMULA 6
- **Calculation of one side of a right triangle knowing the lengths of the hypotenuse and the other side (for the meaning of terms, refer to formula 5).

**Calculate an unknown side 'a' of a right triangle (Figure 1-e)**

** **

**
FORMULA 7
- **Calculation of the surface of a right triangle knowing both sides of the right angle.

**Calculate the area of a right triangle (Figure 1-e)**

** **

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

**Calculate the diagonal of a square (Figure 2-a)**

**
FORMULA 9
- **Calculation of the surface of a square knowing the length of the side.

**Calculate the area of a square knowing the length of the side (Figure 2-a)**

**Example (Figure 2-a) :**

**Data : c** **=**
**50 mm**

**Area : S** **=**
**50 ^{2} =**

**
FORMULA 10
-** Calculation of the surface of a square knowing the length of the diagonal.

**Calculate the area of a square knowing the length of the diagonal (Figure 2-a)**

**S = d ^{2}**

**S = Area**

**d = diagonal**

**Example (Figure 2-a) :**

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

Area **:** **S
70,70 ^{2} /**

__Compare this result with that obtained by applying Formula 9__. The difference of **0.755 mm ^{2}** (

(To facilitate the reading, we report the same figure below to know Figure 2).

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

**Calculate the diagonal of a rectangle (Figure 2-b)**

** **

(This formula above is similar to formula 5).

**
FORMULA 12
-** Calculation of the surface of a rectangle knowing the values of the base and the height.

**Calculate the area of a rectangle (Figure 2-b)**

**S** **=** **b x
h**

**S** **= Area**

**b** **= based**

**h** **= height**

**Example (Figure 2-b) :**

**Data : b =** **10
cm ; h = 5 cm**

**Area : S =** **10 x 5 =** **50 cm ^{2}**

**
FORMULA 13
- **Calculation of the surface of a diamond knowing the length of the diagonals (the diamond is a quadrilateral with four equal sides and unequal adjacent angles).

**Calculate the surface of a diamond (Figure 2-c)**

** S = D x d / 2**

**S** **= Area**

**D = large diagonal**

**d =** **small diagonal**

**Example (figure 2**-**c) :**

**Data : D =** **8 cm ; d = 5 cm**

**Area : S** **=**
**8 x 5 /** **2 = 40 / 2** **=** **20 cm ^{2}**

**
FORMULA 14
-** Calculation of the surface of a parallelogram knowing the values of the base and the height.

**Calculate the area of a parallelogram (Figure 2-d)**

**S = b x h**

**S =** **Area**

**b = based**

**h** **= height**

(This formula above is similar to formula 12).

**Example (figure 2-d) :**

**Data : b = 15 cm ;
h** **= 6 cm**

**Area : S = 15 x 6 =** **90 cm ^{2}**

**
FORMULA 15
-** Calculation of the surface of a trapezium knowing the values of the two bases and the height.

**Calculate the area of a trapeze (Figure 2-e)**

**
FORMULA 16
-** Calculation of the surface of a regular pentagon knowing the length of the sides (the regular pentagon is a polygon with five equal sides and five equal angles).

**Calculate the area of a regular pentagon (Figure 3-a)**

**S
1,72 c ^{2} **

**S = Area**

**c** **= side**

**Example (Figure 3-a) :**

**Data : c =** **20 mm**

**Area : S
1,72 x 20 ^{2} =**

** **

**
****FORMULA 16 - 1 : Regular and irregular polygons**

A polygon is said to be regular when all its sides and angles are congruent (equal).

A polygon is said to be irregular when some of its sides and some of its angles are unequal (incongruous).

**Regular polygon
Polygon irregular**

**
FORMULA 17
-** Calculating the area of a regular hexagon knowing the length of one side (the regular hexagon is a polygon with six equal sides and six equal inner angles).

**Calculate the area of a regular hexagon (Figure 3-b)**

**S =
2,60 x c ^{2 } **

**S =** **Area**

**c =** **Side**

**Example (Figure 3-b "above") :**

**Data : c = 12 mm**

**Area : S
2,60 x 12 ^{2} = 2,60 x 144 = 374,4 mm^{2} **

**
FORMULA 18
- **Calculating the perimeter of a circle (circumference) knowing the value of the diameter.

**Calculate the perimeter of a circle (Figure 3-c)**

**
FORMULA 19
- **Calculation of the surface of a circle knowing the value of the diameter.** **

**Calculate the area of a circle (Figure 3-c)**

** **

**
FORMULA 20
- **Calculation of the length of an arc knowing the value of the angle in the center and the length of the radius.

**Calculate the length of an arc (Figure 3-d)**

** **

(To facilitate the reading, we report the same figure to know figure 3)

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

**Calculate the area of a circular sector (Figure 3-d)**

** **

**
FORMULA 22
-** Calculation of the surface of a circular crown knowing the value of the two diameters (a circular crown is the plane surface between two concentric circumferences).

**Calculate the area of a circular crown (Figure 3-e)**

** **

**
FORMULA 23
-** Calculation of the surface of a parabola segment knowing the value of the base and the height (called the parabola segment the flat surface between a parabola arc and the chord subtended between the ends of the arc).

**Calculate the area of a parabola segment (Figure 4-a)**

**S = 2 /** **3 x b x h**

**S = Area**

**b** **= based**

**h = height**

**Example (Figure 4-a) :**

**Data : b ****=
12 cm ; h = 8 cm**

**Area : S ****= 2 /** **3 x 12 x 8 = 2 / 3 x 96 =** **(2 x 96) /** **3 = 64 cm ^{2}**

** **

**
FORMULA 24
- **Calculation of the surface of an ellipse knowing the length of the two axes.

**Calculate the surface of an ellipse (Figure 4-b)**

** **

**
FORMULA 25**
**-**** **Calculation of the length of a helix knowing the number of turns, the values of the diameter and the height.

**Calculate the length of a helix (Figure 4-c)**

** **

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

**Calculate the volume of a cube (Figure 5-a)**

**V =** **a ^{3 }**

**V** **= volume**

**a =** **bone**

**Example (Figure 5-a) :**

**Data : a =** **4 cm**

**Volume : V = 4 ^{3} =**

** **

**
FORMULA 26 - 1**
**:** Calculation of a diagonal of a cube.

The famous "diagonal of the cube" of edge a is the hypotenuse of a right triangle whose two other sides are **:**

- an edge of the cube (length a)

- a diagonal of a square ABCD forming a face (length b to be determined)

Let's apply the Pythagorean theorem in a triangle (on one side) **ABC**

**a² + a² = b² 2a² = b²
b = a2**

We will now calculate the length **c** of the diagonal of the cube **:**

**a² + b² = c²
a² + 2a² = c²
c² = 3a²
c = a3**

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

**Calculate the volume of a parallelepiped (Figure 5-b)**

**V = a x b x h**

**V = volume**

**a =** **length of the base**

**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 mm ^{3} =**

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

**Calculer le volume d'un cylindre (figure 5-c)**

** **

**FORMULA 28 - 1 : **To calculate a cylinder of a volume generated by the rotation of a rectangle around one of its sides (**lateral surface = 2Rh,** **total surface = 2R (h + R), volume = R ^{2}h, h** being the height and

**
FORMULA 29
- **Calculation of the volume of a hollow cylinder knowing the values of the two diameters and the height.

**
FORMULA 30**
**-** Calculation of the volume of a square section ring knowing the values of the external and internal diameters.

**Calculate the volume of a section ring (Figure 6-a)**

** **

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

**Calculate the volume of a "ring with circular section" torus (Figure 6-b)**

** **

**
FORMULA 32
- **Calculation of the surface of a sphere knowing the value of the diameter.

**Calculate the surface of a sphere (Figure 7-a)**

**Example (Figure 7-a) :**

**Data : d = 15 mm**

**Area : S
3,14 x 15 ^{2} = 3,14 x 225 = 706,5 mm^{2} **

** **

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

**Calculate the volume of a sphere (Figure 7-a)**

** **

**Example (Figure 7-a) :**

**Data : d =** **15 mm**

**Volume : V
0,523 x 15 ^{3} =**

**
FORMULA 34
-** Calculation of the surface of a spherical cap knowing the values of the diameter of the contour and the height.

**Calculate the surface of a spherical cap (Figure 7-b)**

**
FORMULA 35
- **Calculation of the volume of a spherical cap knowing the value of the diameter of the base and the height.

**Calculate the volume of a spherical cap (Figure 7-b)**

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

**Calculate the volume of a dish (Figure 7-c)**

** **

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