How to Calculate the Area of a Rectangle on a Graph

Calculating the area of a rectangle on a graph can seem challenging at first, but with the right approach, it becomes straightforward. The area of any rectangle is determined by multiplying the lengths of two adjacent sides. In this guide, we'll walk through the steps to find these lengths using coordinates on a graph.

First, let’s define our rectangle's corners with labeled points, such as A, B, C, and D. The coordinates for these points are crucial for our calculations. For instance, if point A is at (-8, 2) and point B is at (0, 8), we can find the length of the side connecting these two points (AB) using the distance formula:

$$ ext{distance} = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$

This formula helps us determine the distance between any two points (x1, y1) and (x2, y2) on a Cartesian plane. Let's break it down step-by-step:

### Step 1: Calculate the Length of Side AB
For points A (-8, 2) and B (0, 8):
- The change in x (Δx) is calculated as 0 - (-8) = 8.
- The change in y (Δy) is calculated as 8 - 2 = 6.
Using the distance formula:

$$AB = \\sqrt{8^2 + 6^2} = \\sqrt{64 + 36} = \\sqrt{100} = 10$$

### Step 2: Calculate the Length of Side BC
Next, for points B (0, 8) and C (4, 2):
- The change in x (Δx) is 4 - 0 = 4.
- The change in y (Δy) is 2 - 8 = -6.
Using the distance formula:

$$BC = \\sqrt{4^2 + (-6)^2} = \\sqrt{16 + 36} = \\sqrt{52} = 2\\sqrt{13}$$

### Step 3: Calculate the Area of the Rectangle
Now that we have the lengths of two adjacent sides (AB = 10 and BC = 2√13), we can find the area of the rectangle:

$$ ext{Area} = ext{length} imes ext{width} = 10 imes 2\\sqrt{13} = 20\\sqrt{13} ext{ square units.}$$

### Real-World Applications
Understanding how to calculate the area of a rectangle is not just a theoretical exercise; it has practical applications in various fields such as architecture, engineering, and even landscaping. Whether you're designing a floor plan or determining how much paint is needed for a wall, knowing how to find the area is essential.

In conclusion, remember that the key steps involve identifying the coordinates of your points, applying the distance formula to find the lengths of the sides, and then multiplying to get the area. With practice, calculating areas of rectangles on graphs will become a breeze!