技术问询：求解正方形$OABC$的面积

Alright, let's break down how to find the area of square OABC. Since you haven't shared specific coordinates or measurements for the vertices, I'll cover the most common scenarios and straightforward methods you can apply:

Method 1: Using Side Length (If Known or Calculable) #

The area of a square is simply the square of its side length (Area = s², where s is the length of one side).

• If you know the length of any side (like OA, AB, BC, or CO), just multiply that length by itself to get the area.
• If you have coordinates for two adjacent vertices (e.g., O(0,0) and A(3,0)), calculate the distance between them using the distance formula: s = √[(x₂ - x₁)² + (y₂ - y₁)²], then square the result. For this example, s = 3, so area is 3² = 9.

Method 2: Using Coordinates with the Shoelace Formula #

If you have the coordinates of all four vertices (O, A, B, C), the shoelace formula will give you the area directly. Here's how it works:

1. List the coordinates in order (either clockwise or counter-clockwise):
   Let’s say O(x₀,y₀), A(x₁,y₁), B(x₂,y₂), C(x₃,y₃), then repeat the first coordinate at the end: (x₀,y₀).
2. Apply the formula:

   Area = ½ |(x₀y₁ + x₁y₂ + x₂y₃ + x₃y₀) - (y₀x₁ + y₁x₂ + y₂x₃ + y₃x₀)|
   

   For example, if O(0,0), A(2,1), B(1,3), C(-1,2) (a rotated square), plugging in gives:

   ½ |(0*1 + 2*3 + 1*2 + (-1)*0) - (0*2 + 1*1 + 3*(-1) + 2*0)|
   = ½ |(0 + 6 + 2 + 0) - (0 + 1 -3 + 0)|
   = ½ |8 - (-2)| = ½ *10 = 5
   

   Which is correct, since the side length here is √[(2-0)²+(1-0)²] = √5, so area is (√5)²=5.

Method 3: Using Vector Cross Product (For Coordinate-Based Squares) #

If you have vectors for two adjacent sides (e.g., vector OA and vector OC, since O is often the origin), the area of the square is the absolute value of the cross product of these two vectors. For 2D vectors OA = (x₁,y₁) and OC = (x₂,y₂), the cross product is x₁y₂ - x₂y₁—take its absolute value, and that's your area.

• Example: O(0,0), A(1,2), C(-2,1). Cross product is (1*1) - (-2*2) = 1 +4=5, absolute value is 5, which matches the area of the square with side length √5.

Pick the method that fits the information you have, and you'll get the area in no time!

内容的提问来源于stack exchange，提问作者Marva Jami