How to Count Rectangles on a 5X5 Grid

••• SDI Productions/E+/GettyImages

A 5x5 grid is made up of 25 individual squares, which can be combined to form rectangles. Counting them is a simple matter of adopting a regular approach, which leads to a somewhat surprising result.

    Start with the square in the upper-left corner. Count the number of rectangles which can be created starting with this square. There are five different rectangles with a height of 1, five different rectangles with a height of 2, which leads to 5 x 5, or 25 different rectangles starting with this square.

    Move one square to the right and count rectangles starting here. There are four different rectangles with a height of 1, four more with a height of 2, leading to 5 x 4, or 20 different rectangles starting here.

    Repeat this for the next square over, and you’ll find there are 5 x 3 rectangles, or 15. You should see the pattern by now. For any square, the number of rectangles you can draw is equal to their coordinate distance from the lower right corner.

    Fill out the grid with count of each square’s rectangles, either by manually counting them or by using the trick from step 3. When you’re done, it should look something like this:

    25 20 15 10 5 20 16 12 8 4 15 12 9 6 3 10 8 6 4 2 5 4 3 2 1

    Add up the numbers in the grid to get the total count of rectangles. The answer is 225, which is 5 cubed. Any grid of NxN size will make N cubed rectangles. See the references for the mathematical proof, if you don’t mind a little algebra.

Related Articles

How to Make a Hexagon on Grid Paper
How to Calculate the Square Foot of an Irregular Polygon
How to Convert Square Dimensions to Round
Difference Between Vertices & Edges
How to Find Dimensions in Geometric Shapes
How to Find the Missing Slope of a Trapezoid
How to Measure the Length of the Diagonal Line of a...
How to Calculate the Square Feet on a Cube
How to Find the Dimensions of a Square With the Area
How to Draw an Octagon or 8 sided Polygon
How to Find the Height of a Square
How to Split a Triangle Into Fourths
How to Find the Lateral Area of a Square Pyramid
How to Calculate Area of an Obtuse Triangle
How to calculate the volume of a box
How to Solve a Hexagon
How to Calculate Regression Coefficient
How to Calculate Volume of a Rectangular Prism
How to Make a Grid Map
How to Calculate Diagonal Distance Between Corners...

Dont Go!

We Have More Great Sciencing Articles!