To find the area of a square when you have the coordinates of its vertices, you first need to determine the length of one of its sides. The area of a square is then calculated by squaring its side length.
Calculating Area from Coordinates
A square is defined by four vertices. To find its area, the primary step is to calculate the distance between any two adjacent vertices. This distance gives you the side length of the square.
Step 1: Determine the Side Length
The side length of the square is equivalent to the distance between any two vertices that are connected by an edge. As noted in the reference, "The side length of the square is the distance between the points A and B. (Or any two adjacent vertices)."
You can find this distance using the standard distance formula between two points (x₁, y₁) and (x₂, y₂):
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
- Example: If the distance between two adjacent vertices is calculated to be 22 units, then the side length of the square is 22. As stated in the reference, "Here, this is 22."
Step 2: Calculate the Area
Once you have the side length, finding the area is straightforward. The area of a square is the side length multiplied by itself (squared).
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Formula: Area = side length × side length = (side length)²
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Example: Using the side length from the reference example (22), the area is calculated as follows: "Area is the side length times itself, or 22 x 22 = 484."
- Area = 22 × 22 = 484 square units.
Related Calculation: Perimeter
While the question specifically asks for the area, it's common to also calculate the perimeter. The perimeter of a square is simply four times the length of one side.
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Formula: Perimeter = 4 × side length
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Example: Using the side length of 22 from the reference, the perimeter is: "Perimeter is four times the side length or 4 x 22 = 88."
- Perimeter = 4 × 22 = 88 units.
Summary of Steps
Here’s a quick overview of the process:
- Identify the coordinates of any two adjacent vertices of the square.
- Use the distance formula to calculate the distance between these two points. This result is the side length.
- Square the side length (multiply it by itself) to get the area of the square.
| Step | Description | Calculation Example (Side = 22) |
|---|---|---|
| Find Side Length | Distance between adjacent vertices | 22 |
| Calculate Area | Side length multiplied by itself | 22 × 22 = 484 |
| (Optional) | ||
| Calculate Perimeter | Four times the side length | 4 × 22 = 88 |
