In mathematics, area refers to the size of the region that a two-dimensional figure occupies. The size of a region in higher dimensions is referred to as volume.
It is often possible to find the area of a region bounded by parts of circles and line segments through elementary means. One can find the area of even more complex regions via the use of calculus.
Rectangles are the most basic figures whose area we can study. It makes sense that the area of a rectangle with length and width is simply .
Once we know the area of a rectangle, we can easily find the area of a triangle by just noting that if our triangle has base and height , then the rectangle with length and width has exactly twice as much area as the original triangle. Thus, the area of a triangle is
We can now find the area of any polygon by breaking it up into triangles.
https://youtu.be/51K3uCzntWs?t=842 \\ https://youtu.be/j3QSD5eDpzU
The letters and are frequently used to stand for area. When there are multiple regions under consideration, subscripts are often employed: might be used to denote the areas of particular regions, or . For example, would mean the area of hexagon .
An alternative notation is to use square brackets around the name of the region to denote its area, e.g. for the area of triangle .
Area of a Regular Polygon
The area of any regular polygon can be found as follows:
Inscribe the figure, with sides of length , in a circle and draw a line from two adjacent vertices to the circumcenter. This creates a triangle that is of the total area (consider the regular octagon below as an example).
Drawing the apothem creates two right triangles, each with an angle of at the top vertex. If the polygon has side length , the height of the triangle can be found using trigonometry to be of length .
The area of each triangle is the base times the height, which can also be expressed as and the area of the entire polygon is .
Area of a Triangle
There are many ways to find the area of a triangle. In all of these formulae, will be used to indicate area.
- where is a base and is the altitude of the triangle to that base.
- Heron's formula: , where and are the lengths of the sides and is the semi-perimeter .
- , where is the radius of the incircle and s is the semi-perimeter.
- where and are adjacent sides of the triangle and is the measure of the angle between them.
- , where are the lengths of the sides of the triangle and is the circumradius.
- , where and the triangle has altitudes , , .
Other formulas equivalent to Heron's
These are especially useful when , for :
Also true, but more complex than above
Area of a Quadrilateral
To find the area of most quadrilaterals, you must divide the quadrilateral up into smaller triangles and find the area of each triangle. However, some quadrilaterals have special formulas to find their areas. Again, is the area.
- Kite - where the s represent the lengths of the diagonals of the kite.
- Parallelogram - , where is the base and is the height to that base.
- Trapezoid - , where the s are the parallel sides and is the distance between those bases.
- Rhombus - a special case of a kite and parallelogram, so either formula may be used here.
- Rectangle - , where is the length of the rectangle and is the width. (This is a special case of the formula for a parallelogram where the height and a side happen to coincide.)
- Square - , where is the length of a side.
- Any quadrilateral - , where is the semiperimeter, , , , and are the side lengths, and and are the measures of angles and , respectively.
- Cyclic quadrilateral - where is the semiperimeter and , , , and are the side lengths. (This is a special case of the formula for the area of any quadrilateral; .)