Geometry Fundamentals: Quadrilateral Definitions, Perimeter, and Area of Quadrilaterals

Raute Grundwissen

A quadrilateral is defined specifically as a Raute (rhombus) if and only if it possesses four sides of equal length. This geometric property distinguishes it from other quadrilaterals by ensuring all sides are congruent.

To determine the perimeter of a Raute, the formula used is $U_R = 4 \times a$. This is demonstrated in a provided example where the side length $a$ is equal to $3\,\text{cm}$. Plugging this into the formula results in $U_R = 4 \times 3\,\text{cm}$, which equals a total perimeter of $U_B = 12\,\text{cm}$.

To determine the surface area of a Raute, the formula is $A_R = (e \times f) : 2$, where $e$ and $f$ represent the lengths of the diagonals. In the provided example, diagonal $e$ is $3.8\,\text{cm}$ and diagonal $f$ is $4.6\,\text{cm}$. The calculation proceeds as follows: $A_R = (3.8\,\text{cm} \times 4.6\,\text{cm}) : 2$. This results in a product of $A_a = (17.48) : 2$, leading to a final area measurement of $A_R = 8.74\,\text{cm}^2$.

Drachen Grundwissen

A quadrilateral is classified as a Drachen (kite) when it possesses two pairs of adjacent sides that are equal in length. This specific arrangement of sides defines the symmetry of the shape.

To calculate the perimeter of a Drachen, the formula is $U_D = 2 \times (a + b)$. For example, if the side lengths are given as $a = 6\,\text{cm}$ and $b = 3\,\text{cm}$, the calculation is formatted as $U_D = 2 \times (6\,\text{cm} + 3\,\text{cm})$. Adding the terms inside the parentheses gives $U_D = 2 \times (9\,\text{cm})$, resulting in a perimeter of $U_p = 18\,\text{cm}$.

To calculate the area of a Drachen, the formula is $A_D = (e \times f) : 2$. In a practical example where diagonal $e = 7\,\text{dm}$ and diagonal $f = 8\,\text{dm}$, the calculation is $A_D = (7\,\text{dm} \times 8\,\text{dm}) : 2$. This results in $A_p = (56\,\text{dm}) : 2$, which gives a final area of $A_p = 28\,\text{dm}^2$.

Trapez Grundwissen

A quadrilateral is recognized as a Trapez (trapezoid) when two of its opposite sides are parallel to each other. This is the fundamental requirement for this specific shape.

To calculate the perimeter of a Trapez, the formula is $U_T = a + b + c + d$, where all four side lengths are summed together. In the example provided, the sides are given as $a = 6\,\text{cm}$, $b = 3\,\text{cm}$, $c = 76\,\text{cm}$, and $d = 9\,\text{cm}$. The calculation is written as $U_p = 6\,\text{cm} + 3\,\text{cm} + 76\,\text{cm} + 9\,\text{cm}$, which results in a perimeter value of $U_T = 34\,\text{cm}$.

To calculate the area of a Trapez, the formula used is $A_T = (a + c) \times h : 2$. In the example calculation, the values used are $a = 0.7\,\text{dm}$, $c = 24\,\text{dm}$, and height $h = 32\,\text{dm}$. The process is shown as $A_+ = (0.7\,\text{dm} + 24\,\text{dm}) \times 32\,\text{dm} : 2$. The intermediate step results in $A_F = (24.7\,\text{dm}) \times 32 : 2$. The document records the product as $A_r = 777.3 : 2$, resulting in a final area of $A_+ = 385.6\,\text{dm}^2$.

Parallelogramm Grundwissen

A quadrilateral is called a Parallelogramm (parallelogram) only if the opposite sides are parallel to each other. When performing calculations for this shape, it is essential to first determine and ensure that the side lengths are expressed in the same unit of length.

To calculate the perimeter of a Parallelogramm, the formula used is $U_P = 2 \times (a + b)$. In the specific example provided, the side lengths are $a = 8\,\text{cm}$ and $b = 9\,\text{cm}$. The calculation is carried out as $U_P = 2 \times (8\,\text{cm} + 9\,\text{cm})$, then $U_P = 2 \times (17\,\text{cm})$, resulting in a total perimeter of $U_p = 34\,\text{cm}$.

To calculate the area of a Parallelogramm, the formula is $A_P = a \times h_a$, where $h_a$ is the height corresponding to side $a$. In the example, side $a = 7\,\text{dm}$ and height $h_a = 18\,\text{dm}$. The resulting calculation is $A_P = 7\,\text{dm} \times 18\,\text{dm}$, totaling an area of $A_P = 126\,\text{dm}^2$.