How to calculate the area of ​​a parallelogram: 11 steps (with pictures)

How to calculate the area of ​​a parallelogram: 11 steps (with pictures)
How to calculate the area of ​​a parallelogram: 11 steps (with pictures)
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A parallelogram is a geometric figure made up of four sides, two by two parallel. Squares, rhombuses, and rectangles are special cases of parallelograms, although most people, when they think of a parallelogram, imagine the classic figure of a rectangle with two flat sides and two diagonal sides. Regardless of the inclination of a parallelogram or the angle of vertices, calculating area is an extremely simple procedure.

Steps

Part 1 of 2: Find the area of ​​a parallelogram

Calculate the Area of ​​a Parallelogram Step 1

Step 1. Multiply the base by the height

If the geometry problem you have to deal with provides the measurement of the base and height of the figure in question, you will only need to multiply them to get the area. Suppose the base of your figure is 5 cm and the height is 3 cm, the area will be equal to 15cm2 { displaystyle 15 \, cm ^ {2}}

{displaystyle 15\, cm^{2}}
, donc 5∗3=15{displaystyle 5*3=15}

5*3=15
  • La base d’un parallélogramme correspond au côté le plus long sur lequel repose la figure.
  • Alors que la hauteur représente la distance minimale qui sépare les deux côtés les plus longs de la figure.
  • C’est à vous de choisir le côté à utiliser comme la base et la hauteur. En fait, vous pouvez pivoter un parallélogramme autant que vous voulez pour le poser sur le côté souhaité, mais le résultat final sera toujours le même.
Calculate the Area of ​​a Parallelogram Step 2

Step 2. Measure and write down the length of the side that will be the base

A parallelogram is made up of two pairs of sides parallel to each other. Normally, the figure rests on one of the four sides, which gives the pair of segments in question a perfectly flat appearance. Measure the length of one of these two sides and consider it as the base or just write B.

For example, suppose that the base of the parallelogram studied is 10 cm long

Calculate the Area of ​​a Parallelogram Step 3

Step 3. Draw a straight line from the base to its parallel side

The line in question must form an angle of 90 ° with the base so that the height measurement is perpendicular to it. The easiest way to do this is to use a ruler and measure the minimum distance, in a straight line, between one of the bottom corners of the figure and the opposite side.

It is important not to make the mistake of considering one of the oblique sides as the height of the parallelogram

Calculate the Area of ​​a Parallelogram Step 4

Step 4. Find the height

You just need to measure the distance from the base to the opposite side. As long as the line you are measuring is perpendicular to the base of the figure (that is, it forms a 90 ° angle with it), you will be sure to get the height measurement. Represent it by the letter H.

  • In our example, we assume that the parallelogram in question has a height of 5 cm.
  • The line indicating the height can also be drawn on the outside of the figure.
Calculate the Area of ​​a Parallelogram Step 5

Step 5. Multiply the base by the height to find the area

After obtaining the necessary data, simply enter them into the following equation: A = B ∗ H { displaystyle A = B * H}

{displaystyle A=B*H}

, où A est l’aire de la figure. En effectuant les calculs, nous obtiendrons ce qui suit:

  • A=B∗H{displaystyle A=B*H}
  • {displaystyle A=B*H}

    ;

    • B=10cm;H=5cm{displaystyle B=10\, cm\,;H=5\, cm}
    • {displaystyle B=10\, cm\,;H=5\, cm}

      ;

  • A=10cm∗5cm{displaystyle A=10\, cm*5\, cm}
  • {displaystyle A=10\, cm*5\, cm}

    ;

  • L’aire de notre parallélogramme=50cm2{displaystyle =50\, cm^{2}}
  • {displaystyle =50\, cm^{2}} .
Calculate the Area of ​​a Parallelogram Step 6

Step 6. Know that the area is expressed in square units

In our example, the final result is simply 50. In reality, this formulation is incomplete, because it does not indicate the real dimension of the studied figure (in fact, we do not know if it is centimeters, of meters, inches, etc.). Since area is a measure of space, it is also necessary to indicate the size of this space to the reader, teacher or client. Since the examples given in this article are expressed in centimeters, the end result should be expressed in square centimeters. This means that our parallelogram is made up of 5 perfect squares one centimeter apart.

  • To express the end result of the problem in its correct form, write the character ² at the top of the unit of measure in which the initial data is expressed. For example, if the base and height are expressed in meters, the final result should be expressed in square meters or m2 { displaystyle m ^ {2}}

    m^{2}
  • Si les données initiales ne sont accompagnées d’aucune unité de mesure, le résultat final est suivi du mot unité².

Partie 2 sur 2: Calculer l’aire d’un parallélépipède

Calculate the Area of ​​a Parallelogram Step 7

Step 1. Proceed like any other flat figure problem

Parallelepipeds are nothing more than three-dimensional parallelograms. Calculating the area of ​​these solids is very simple. It is enough to know their length (L), their height (H) and their width (l) and to insert them in the following mathematical formula:

  • Side surface = 2 (LH + Ll + Hl) { displaystyle 2 (LH + Ll + Hl)}

    2(LH+Ll+Hl)
    .
Calculate the Area of ​​a Parallelogram Step 8

Step 2. Identify the length and height of one of the faces

If you are studying a rectangular solid (a box, for example), one of whose faces is a parallelogram, you can measure length and height just as you measure the height and length of a parallelogram in a plane. Remember that for the measurements to be correct, the lines representing these two quantities must be perpendicular, that is, they must form an angle of 90 °. Next, write down the data obtained as the length and height of the solid, respectively.

  • In the case of a parallelogram, it is good to remember that the height is not the measurement of the oblique side, but the minimum distance, in a straight line, which separates the side identified as the base from the opposite and parallel side.
  • In our example, suppose we have a parallelepiped with these dimensions L = 6; H = 4 { displaystyle L = 6 \,; H = 4}

    {displaystyle L=6\,;H=4}
    , exprimées en cm.
Calculate the Area of ​​a Parallelogram Step 9

Step 3. Find the width

To do this, measure the side going in the opposite direction to that used for length and height. This is the last dimension you haven't measured yet. Just make sure that you are not measuring any of the already known sides and that they are parallel to those used to get the length or height of the solid in question. From the same point of origin (often one of the vertices of the parallelepiped), you should be able to measure the three quantities in question and each side should be perpendicular to the other two.

  • In our example, we assume that the solid in question has a width equal to l = 5cm { displaystyle l = 5 \, cm}

    {displaystyle l=5\, cm}
    .
Calculate the Area of ​​a Parallelogram Step 10

Step 4. Enter the measured data into the starting formula

If all three sizes (length, width and height) are already provided in the problem or if you have successfully measured them, you can get the final result. Enter the data in your possession in the following formula:

  • Side surface = 2 (LH + Ll + Hl) { displaystyle 2 (LH + Ll + Hl)}

    2(LH+Ll+Hl)
    ;

    • L=6cm;H=4cm;L=5cm{displaystyle L=6\, cm\,;H=4\, cm\,;L=5\, cm}
    • {displaystyle L=6\, cm\,;H=4\, cm\,;L=5\, cm}

      ;

  • Surface latérale=2(6∗4+6∗5+4∗5){displaystyle =2(6*4+6*5+4*5)}
  • =2(6*4+6*5+4*5)

    ;

  • Surface latérale=2(24+30+20){displaystyle =2(24+30+20)}
  • =2(24+30+20)

    ;

  • Surface latérale=2(74){displaystyle =2(74)}
  • =2(74)

    ;

  • Surface latérale=148cm2{displaystyle =148\, cm^{2}}
  • {displaystyle =148\, cm^{2}} .
Calculate the Area of ​​a Parallelogram Step 11

Step 5. Know that the area is expressed in square units

In our example, the end result, if it's just shown as 148, doesn't make sense, because we don't know if it's referring to kilometers, centimeters, meters, inches, etc. Since area is obviously another form of surface, it is necessary to express the result in squared units, even if you are measuring three-dimensional geometric objects or figures. For our example, the data will be expressed in square centimeters.

  • If during the problem you forget which unit of measurement to use, always refer to the original data provided to you or that you measured. Remember that this expression 32 { displaystyle 3 ^ {2}}

    3^{2}

    correspond à l’opération mathématique suivante 3∗3{displaystyle 3*3}

    3*3

    . pour calculer l’aire d’une surface, il suffit de multiplier les mesures relatives, par exemple a=3m∗3m{displaystyle a=3m*3m}

    {displaystyle a=3m*3m}

    . dans cet exemple, vous pouvez simplement indiquer que la surface est égale à 32{displaystyle 3^{2}}

    3^{2}

    , mais aussi préciser que l’unité de mesure est m2{displaystyle m^{2}}

    m^{2}

    ..

conseils

pour tester vos compétences et avoir la preuve de la validité d’un des théorèmes d’euclide, tracez une diagonale reliant deux angles opposés du parallélogramme, puis tracez deux lignes perpendiculaires entre elles et parallèles aux côtés de la figure géométrique, dont le point d’intersection tombe sur la diagonale. en observant le dessin obtenu, vous remarquerez que deux parallélogrammes ont été formés à l’intérieur du premier, opposés l’un à l’autre et non coupés par la diagonale de ce dernier. la démonstration consiste à montrer que, quel que soit le tracé des lignes perpendiculaires, les deux parallélogrammes obtenus auront toujours la même surface.

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