From the Cartesian Plane to the Areas of Rectangles and Triangles

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"René Descartes (original by Frans Hals in public domain)" by Rafael Robles L.

“René Descartes” portrait by Frans Hals

This article deals with calculating the areas of squares and triangles. It is an easy introduction to a series eventually leading to calculus, the mathematics of Newton and Leibniz.

But let’s not put the cart before the horse. On second thought, we do need to start with Descartes.

The Cartesian Plane

René Descartes, a French citizen who died in Sweden, lived from 1596 to 1650. He developed the “Cartesian plane”, the familiar X-Y coordinate system mapped onto a flat plane. This is a blend of geometry and algebra, and will become very useful later in this series of articles.

The Cartesian plane has a horizontal X-axis and a vertical Y-axis. Any point in the plane has an X-Y coordinate value such as (x=1, y=1). Usually this is shown as (1, 1). To discuss any arbitrary set of points, notation such as “(x1, y1), (x2, y2), (x3, y3)…” is used.

The plane is usually drawn with the X-axis and Y-axis meeting in the centre, at coordinate (0,0). However, this article will focus on the first quadrant, where both X > 0 and Y > 0.

This next image shows a two-tone square, with each side 6 units long. The corners are at (1, 1), (1, 7), (7, 7) and (7, 1).

"Simple Graph in the Cartesian Plane" by Mike DeHaan

“Simple Graph in the Cartesian Plane” Image by Mike DeHaan

The length of any line from (x1, y1) to (x2, y2) is the square root of the sum of the squared difference in the “X” direction and the squared difference in the “Y” direction. Algebraically, this is L = square root( (x2 – x1)^2 + (y2 – y1)^2 ).

Finding the length of a vertical or horizontal line, such as the bottom line labelled “base” or the right-hand “height”, is much simpler. Either the change in “X” or in “Y” is zero, and the square root of a squared number is the original number. So the length of a vertical line is simply (y2 – y1); of a horizontal line, (x2 – x1).

In the above image, the length of each side is (7 – 1) = 6.

Yes, it is tempting to look at the image and say “7”…but the starting point for the line is (1, 1), not (0, 0).

The Area of a Rectangle

Hopefully we all agree that the area of a rectangle is the length of one side times the length of either neighbouring side. In the image to the left, these are labelled “Base” and “Height”.

"Area of a Diagonalized Square" by Mike DeHaan

“Area of a Diagonalized Square” Image by Mike DeHaan

This article happens to use a square, but of course a square is simply a special type of rectangle. Readers are invited to set up their own graphs, using their favorite spreadsheet programs or graph paper, to see how non-square rectangles might appear.

The mathematical formulation would be A = h * b, with ‘A’ for “area”, ‘h’ for “height” and ‘b’ for “base”. In this image, the area of the square is 6 * 6 = 36.

The Area of a Triangle

This image also bisects the rectangle on the diagonal, using the heavy blue line that originally allowed Excel to build the graph. Visually, it is obvious that the two equal triangles cover the same area as the square. So each triangle, the purple and the green, has an area of 36/2 = 18. In general, the area of a triangle is “one-half of the base times the height”, or A = (b * h)/2.

The Importance of Area in the Cartesian Plane

The diagonalized square was simply an introduction to determining the areas of rectangles and triangles, as well as an elegant way to generate a graph in Excel. A later article may return to this image to help introduce differential calculus.

The subject of integral calculus relies heavily on the ability to calculate the areas of rectangles in the Cartesian plane.

This article is reaching its limits, but those articles on calculus will push mathematics to its limits…their limits as changes approach zero.

References:

Jay Abramson, Terry Turner and the ASU Department of Mathematics and Statistics, “The Cartesian Plane“, 2004, referenced July 16, 2011.
Marta Hidegkuti, City Colleges of Chicago, “Notes on Area”, PDF referenced July 16, 2011.

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