The Definition of Area in Mathematics
Area is a dimensionless real number that measures the surface area of a flat shape, such as a polygon, circle or ellipse. It is an important concept in mathematics because it enables us to calculate the size of shapes and surfaces. It is a useful tool to help children understand the concept of space and also helps them to recognize and measure the size and volume of different objects they encounter in their day-to-day lives.
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The concept of area was developed in Antiquity. It was first determined on the class of polygons (figures that can be decomposed into a finite number of triangles without common interior points). In this class, an area with the properties 1)-4) exists and is unique [1].
There are many ways to find the area of a figure, for example by multiplying its length by its width, finding its perimeter or by using the method of dissection. These methods can be used to calculate the area of a wide variety of shapes, from circles and parallelograms to cubes and pyramids.
A square, for instance, has a width and a length, which is why it is easy to find its area. A circle, on the other hand, has a radius and a height, which makes it more difficult to find its area.
In the early days of geometry, areas were only defined in terms of other areas. For instance, a square had twice the area of two congruent right-angled isosceles triangles put together, and a rectangle with sides of integer length aa and bb had abab times that of the square.
Eventually, people started to realize that they could use the same area formula to calculate the area of other shapes. The ancients had always thought that the area of a circle was equal to its perimeter, but this was not true. They also found that the area of a rectangle with irrational sides was twice that of its square.
Once they had this new theory, it was possible to evaluate the area of other figures more rigorously. Rather than simply dividing a circle into triangles, they could now divide it into rectangles – and so the concept of area was born!
The standard unit of area is the square meter. A square meter is the area of a square with side lengths one meter. It is also sometimes written as m2.
There are several other standard units of measurement for the area of geometric shapes. These include square centimeters, square meters and square feet.
What is the difference between perimeter and area?
Area and perimeter are different because they measure two different aspects of a shape. The area of a shape is the total surface area, whereas the perimeter of a shape is just a number that shows the distance along the edges.
Often, in real life, shapes can be more complex and require a different way of calculating the area. This is why it is often helpful to show children that the area of a shape can be found by dividing it into smaller parts or by adding their areas together.
In conclusion, the concept of area is a fundamental concept in mathematics that allows us to measure the size of flat shapes and surfaces. It has practical applications in various fields, from geometry and architecture to everyday life situations. The concept of area dates back to antiquity, where it was initially defined for polygons and later extended to other shapes. Various methods, such as multiplying length and width or using the method of dissection, can be employed to calculate the area of different shapes. The development of the area formula revolutionized the field of geometry and enabled more rigorous evaluation of areas. The difference between area and perimeter lies in their measurements, with area representing the total surface area and perimeter indicating the distance along the edges. Understanding the concept of area is important for children as it helps them comprehend spatial relationships and enables them to recognize and measure the size and volume of objects in their surroundings. By breaking down complex shapes into smaller parts, children can grasp the concept of area more effectively. Standard units, such as square meters, square centimeters, and square feet, are used to measure area, providing a standardized means of comparison and communication. Overall, the concept of area plays a vital role in mathematics and everyday life, allowing us to quantify and understand the sizes and dimensions of various shapes and surfaces.