Just as a segment can be measured by comparing it with a unit segment, the area of a polygon or other plane figure can be measured by comparing it with a unit square. The common formulas for calculating areas reduce this kind of measurement to the measurement of certain suitable lengths. The simplest case is a rectangle with sides a and b, which has area ab. By putting a triangle into an appropriate rectangle (see figure![[Credits : Encyclopædia Britannica, Inc.]](http://media-2.web.britannica.com/eb-media/49/70549-003-F9943B4A.gif)
), one can show that the area of the triangle is half the product of the length of one of its bases and its corresponding height—bh/2. One can then compute the area of a general polygon by dissecting it into triangular regions. If a triangle (or more general figure) has area A, a similar triangle (or figure) with a scaling factor of s will have an area of s2A.
The-figure-illustrates-the-three-basic-theorems-that-triangles-areThe figure illustrates the three basic theorems that triangles are congruent (of equal shape and …[Credits : Encyclopædia Britannica, Inc.]
Proof-that-the-sum-of-the-angles-in-a-triangleProof that the sum of the angles in a triangle is 180 degrees.[Credits : Encyclopædia Britannica, Inc.]
The-formula-in-the-figure-reads-k-is-to-lThe formula in the figure reads k is to l as m is to n if and only if …[Credits : Encyclopædia Britannica, Inc.]
[Credits : Encyclopædia Britannica, Inc.]
Thales-of-Miletus-is-generally-credited-with-giving-the-firstThales of Miletus (fl. c. 600 bc) is generally credited with giving the first proof that for any …[Credits : Encyclopædia Britannica, Inc.]
[Credits : Encyclopædia Britannica, Inc.]
Bridge-of-AssesBridge of Asses.[Credits : Encyclopædia Britannica, Inc.]
Quadrilateral-of-Omar-Khayyam-Omar-Khayyam-constructed-the-quadrilateral-shownQuadrilateral of Omar Khayyam[Credits : Encyclopædia Britannica, Inc.]
Link to this article and share the full text with the readers of your Web site or blog-post.
If you think a reference to this article on "Euclidean geometry" will enhance your Web site,
blog-post, or any other web-content, then feel free to link to this article,
and your readers will gain full access to the full article, even if they do not subscribe to our service.
You may want to use the HTML code fragment provided below.
We welcome your comments. Any revisions or updates suggested for this article will be reviewed by our editorial staff. Contact us here.
Regular users of Britannica may notice that this comments feature is less robust than in the past. This is only temporary, while we make the transition to a dramatically new and richer site. The functionality of the system will be restored soon.