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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.