Counting shapes

Hello, and welcome to Myriora Maths! How many rectangles can you see in this strip? Most people say four. There are lots more!

Count them by size. There are four rectangles that are one square long. Three that are two squares long. Two that are three squares long. And one that's four squares long: the whole strip. Four plus three plus two plus one is ten!

Here's a quicker way. The strip has five upright lines. Every rectangle uses two of them: one for its left side, and one for its right side. So count the ways to pick two lines out of five: four plus three plus two plus one, which is ten. It's just like the handshakes puzzle!

Now a grid, two squares tall and three squares wide. For the sides, pick two of the four upright lines. That's six ways. For the top and bottom, pick two of the three flat lines. That's three ways. Every pair of sides works with every top and bottom, so we multiply. Six times three is eighteen rectangles!

Triangles can work the same way. Here's a triangle cut into four slices from the top. Every triangle uses two of the five lines from the top corner. So again, there are ten triangles!

A common mistake is counting only the smallest shapes. Bigger rectangles are made of smaller ones, joined together! Count every size, or count the pairs of lines.

Your turn! How many rectangles are there in a strip of five squares? You've got twenty seconds, or pause the video if you need more time. Go!

Time's up! There are six upright lines, so count the pairs: five plus four plus three plus two plus one. That's fifteen rectangles!

Quick review. Count the shapes size by size, from the smallest to the biggest. Or count the ways to pick their sides. In a grid, multiply the side choices by the top and bottom choices. See you next time!