The study of math is incomplete without learning geometry. In geometry, we study about shapes. Every object has a shape. Such as the book you read from, or the cell phone screen you watch videos in. We study the sizes and other features or dimensions of these shapes, in geometry. We do this by figuring out how long, how wide, how spacious they are, etc.
It is better to use examples because math is a real-world subject. Have you ever wondered, what is the length of the Eiffel Tower? How high are the pillars of your school? What could be the width of the road in front of it? How many things can be stored inside your basement room, or how much water can be filled in your bottle? Such properties about things need to be known before they are made. Thus, we measure these different properties or dimensions of the objects. And, these kinds of riddles and problems in math or geometry are called problems of measurements.
We have to measure many things in life! Before making our buying your house, you also had to measure what shape and size your house should have. And by knowing this, we can find out how much room or space is there inside your house! Some of the most common shapes you must have already studied are squares, rectangles, triangles and circles. Here you will see how to find the area inside a rectangle, with the help of a graph.
Before we proceed, let’s clear some concepts that you must already be knowing!
A rectangle is a type of polygon, which is a figure with more than three sides or lines meeting each other at certain angles. Squares and rectangles are quadrilaterals or four sides figures in a two-dimensional plane. In a rectangle, each side is connected to the other -- it’s adjacent side -- at a point called a vertex (vertices, in plural). Any two adjacent sides make a right angle. So there are four angles, each being ninety degrees.
The opposite sides are parallel to each other. But, unlike a square, in a rectangle, one pair of sides is longer than the other. The longer side is called the ‘length’ of the rectangle. The shorter side is called the ‘breadth’ or ‘width’ of the rectangle. Your cell phone or computer screen has a rectangular shape.
Formula for Calculating the Area of a Rectangle
The formula for calculating the area of a rectangle is simple! You have to multiply its length (l) with its breadth (b):
Area = l x b
In some textbooks or math problems, instead of calculating length into breadth, if we call the length as ‘height’ and breadth as ‘base’, then we get:
Area = height x base
Both these formulas are the same.
Definition of Area
But what do we mean by ‘area’? What is the area of a shape like a rectangle in a two-dimensional plane? It is the amount of space contained within the lines that make the rectangle.
But when we are calculating the area on a graph, there will be a little change in the way the area is measured. We will see that soon!
You have seen a graph before. Recall what a graph is made of. There are straight lines running from the top of the page to the bottom, and from side to side. These lines intersect each other and are perpendicular to each other. And they are each 1 cm apart from the other lines on either side. Because of this, many boxes or squares are formed throughout the graph sheet, which is all equal to each other. They all have a side of 1 cm. This is a line graph.
Hence, when a geometric figure like a rectangle is formed on the line graph, we calculate its area in a different way.
Area of a Rectangle on a Graph
Now, let us make this exercise more practical. You can either take a graph paper or simply create a line graph on a blank sheet of paper with lines that are a centimeter apart from each other. You may even use an online geometry site for this exercise.
Ready? Let’s go! Take a point on the graph, where two of the lines intersect. By the way, these points are called coordinates. You have chosen your first coordinate. Now, draw a line. For this example, let’s connect this coordinate point to a point 6 squares to the right or to the left, whichever side you choose. This line will be 6 cm since every square has a side that is 1 cm in size.
Now, connect one of the endpoints of this line to the point that is four squares below this it. Do the same for the other endpoint. The two new lines formed are both 4 cm long. Finally, connect the ends of these two new lines. Another 6 cm long line is created that is parallel to the first line. And, you have created a rectangle on the graph!
There are four coordinates of the rectangle. If you see, some many boxes or squares are contained inside this rectangle. These will help to calculate the area of this rectangle.
Remember the formula for calculating the area of a rectangle, form above?
We know two sides are 6 cm long, and the other two sides are 4 cm long. Thus, applying the formula:
Area = 6 x 4 cm. That is 24 square cm.
Why do we measure the area in “square cm”?
Number of squares in the figure
Now, if we are to calculate the rectangle’s area on the graph, we just have to count the number of squares in the graph. The number of these squares will be the square area or surface area of the rectangle.
If we have not been given the measurement in metre or centimetre, then we state it as “square units”, because each square is a unit measure on the graph.
If you have made the figure, quickly count the number of squares within. Did you do it? Did you find 24 squares within the rectangle? You certainly will have! You can also see that there are 6 columns and 4 rows. Multiplying six with four gives us twenty-four. This way, its area checks out.
Each of these squares is 1 cm in the area themselves, and the area of a square – if you remember – is found by multiplying one side with another side. In this case, the area of each square is 1 cm x 1 cm = 1 sq. cm. Hence, the area of the rectangle is 24 square cm, also written as 24 cm2.
Let’s recap, quickly, what we just learned.
If you have been given a geometrical problem where you have to measure the area of a rectangle with the help of a graph, or on a graph:
- Take 4 points on the graph -- the coordinate points that will be the vertices of the rectangle.
- If the problem says: one pair of opposite sides are 6 units or cms, and the other pair of opposite sides are 4 units or cms.
- Construct the figure as per the measurements.
- Take one point.
- Go six grids to the right to make the line for length, and go four grids down to make the line for breadth.
- Draw the rectangular shape.
The area for the figure is measured on the graph by counting the squares or grids within the rectangle. You will find twenty-four squares within the figure. Hence, the area is twenty-four square units or cms.
You can check this by multiplying length into breadth, that is six times four! You also will find 6 columns and 4 rows. So, the resulting Area is 6 x 4 = 24 sq. cm or 24 cm2
You can see how easy it can be to measure the area of a rectangle on a graph. Math doesn’t have to be tough; it can be fun and challenging. This was only one of the many measurement tips and tricks used in math. There are other measurements in geometry like volume, perimeter, etc., that you will learn soon.
You can head over to Cuemath to discover and to learn how to use more such hacks. These hacks make algebra, geometry, and programming all the more interesting to learn, and make learning of tough subjects far better.