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## Area axioms of figures ☰

The area axioms are a set of fundamental principles that describe how the concept of area behaves in
mathematics. These axioms provide the basis for the measurement of the size of two-dimensional figures
such as squares, circles, and triangles.

The following are the common area axioms of figures:

**Non-Negativity:** The area of any figure is a non-negative real number, meaning it
cannot be negative.
**Additivity:** The area of a union of two non-overlapping figures is equal to the sum
of the areas of the individual figures. In other words, if two figures do not overlap, the total
area of the two figures combined is equal to the sum of their individual areas.
**Homogeneity:** If a figure is scaled by a factor of \(k\), then its area is
multiplied by \(k^2\). For example, if the side of a square is multiplied by 2, then the area of the
square will be multiplied by 4.
**Dimensionality:** The area of a figure is a quantity with dimension of length
squared. For example, if the unit of length is meters, then the unit of area is square meters.

These axioms provide a solid foundation for the study and measurement of area in mathematics. They allow
mathematicians to reason about the properties of figures, to compare the sizes of different figures, and
to make accurate calculations involving area.

## Area of a parallelogram ☰

To calculate the area of a parallelogram, you need to know the length of its base and the height (or
altitude) of the parallelogram. The base is one of the sides of the parallelogram that is perpendicular
to the height. The height is the perpendicular distance from the base to the opposite side of the
parallelogram.

The formula for calculating the area of a parallelogram is: \( \text{Area}=
\text{base} \cdot \text{height} \).

In mathematical notation, this can be written as: \(A=b \cdot
h\), where \(A\) represents the area of the parallelogram, \(b\) represents the length of the base, and
\(h\) represents the height of the parallelogram.

**It is important to note** that the
height of the parallelogram can be drawn from any one of the parallel sides to the opposite side.
Therefore, if you know the length of any of the two parallel sides of the parallelogram and the
perpendicular distance between them, you can calculate the area of the parallelogram by multiplying the
length of the base by the corresponding height.

## Area of a triangle ☰

A triangle is a two-dimensional geometric shape with three sides and three angles. The area of a
triangle is the measure of the surface enclosed by its three sides.

There are different ways to
calculate the area of a triangle, depending on the given information. The most common formula for
calculating the area of a triangle is:

\(A=\frac{1}{2}\cdot \text{base} \cdot \text{height} \) ,
where the base is the length of one of the sides of the triangle, and the height is the perpendicular
distance from the base to the opposite vertex.

To apply this formula, we need to know the length of
the base and the height of the triangle. If the height is not given, it can be calculated using the
Pythagorean theorem, which states that the square of the hypotenuse (the longest side) of a right-angled
triangle is equal to the sum of the squares of the other two sides. Therefore, if we know the lengths of
two sides of a right-angled triangle, we can calculate the length of the third side, which is the height
of the triangle.

If the triangle is not right-angled, we can still calculate the area using the
formula above, provided that we know the length of the base and the height. The height can be found by
drawing a perpendicular line from the opposite vertex to the base.

Another way to calculate the
area of a triangle is to use Heron's formula, which is based on the lengths of the three sides of the
triangle:

\( \text{Area} = \sqrt{ s(s-a)(s-b)(s-c) } \), where \(a\), \(b\) and \(c\) are the
lengths of the three sides of the triangle, and \(s\) is the semiperimeter, which is half the perimeter
of the triangle: \( s = \frac{a+b+c}{2} \).

Heron's formula is useful when the lengths of the sides
are known, but the height is not easily calculable.

In summary, the area of a triangle can be
calculated using the formula \(A=\frac{1}{2}\cdot \text{base} \cdot \text{height} \) , where the base is
the length of one of the sides of the triangle, and the height is the perpendicular distance from the
base to the opposite vertex. Alternatively, the area can be calculated using Heron's formula, which is
based on the lengths of the three sides of the triangle.

## Area of trapezium ☰

A trapezium (or trapezoid) is a four-sided geometric shape with two parallel sides and two non-parallel
sides.The area of a trapezium is the measure of the surface enclosed by its four sides. To calculate the
area of a trapezium, you need to know the length of its parallel sides (the bases) and the perpendicular
distance (the height) between them. The formula for calculating the area of a trapezium is:

\(A=\frac{1}{2}\cdot (a + b)\cdot h\), where \(a\) and \(b\) are the lengths of the parallel sides (the
bases) and h is the height of the trapezium.

**It is important to note** that the
height of the trapezium is the perpendicular distance between the two parallel sides. If the trapezium
is not given with its height, it can be calculated by drawing a perpendicular line from one of the
non-parallel sides to the opposite parallel side.

## Area of rhombus ☰

A rhombus is a two-dimensional geometric shape with four equal sides and opposite angles that are equal.
The area of a rhombus is the measure of the surface enclosed by its four sides.

To calculate the
area of a rhombus, you need to know the length of one of its diagonals. The formula for calculating the
area of a rhombus is:

\( A=\frac{d_1\cdot d_2}{2}\), where \(d_1\) and \(d_2\) are the lengths of
the diagonals of the rhombus.

**It is important to note** that the diagonals of a
rhombus are perpendicular to each other, and they bisect each other. Therefore, the length of each
diagonal is half the product of the lengths of the other diagonal.

In other words: \(d_1=2h_1\) and
\(d_2=2h_2\), where \(h_1\) and \(h_2\) are the lengths of the altitudes (perpendicular heights) of the
rhombus.

Therefore, we can also calculate the area of a rhombus using the lengths of its sides and
one of its altitudes. The formula for this is:

\( \text{Area}=\text{base} \cdot \text{height} \)
where the base is the length of one of the sides of the rhombus, and the height is the perpendicular
distance between the two parallel sides.

In summary, the area of a rhombus can be calculated
using the formula \( A=\frac{d_1\cdot d_2}{2}\) , where \(d_1\) and \(d_2\) are the lengths of the
diagonals of the rhombus. Alternatively, the area can be calculated using the formula \(
\text{Area}=\text{base} \cdot \text{height} \) , where the base is the length of one of the sides of the
rhombus, and the height is the perpendicular distance between the two parallel sides.