当正方形的边增加三倍时，正方形的面积会发生什么变化？

正方形是规则的四边形。它有四个长度相等的边。除此之外，所有四个角度在本质上也是相等的。正方形的角对着一个等于 90 度的角。正方形的对角线相等，并以 90 度角平分。正方形是矩形的一种特殊情况，其中相对的一对边的长度相等。

正方形的性质

正方形具有以下重要属性：

• 一个正方形有4个顶点和4条边
• 正方形的对边相互平行
• 对角线彼此相等。
• 正方形的对角线以 90° 平分
• 对角线的长度大于正方形的对应边
• 所有的内角均等于 90°。

当正方形的边增加三倍时，正方形的面积会发生什么变化？

解决方案：

Since, we know,

Area of a square = (side)²

Let us assume the side of the square to be x and the area to be denoted by A. 

On substituting the values, we get, 

A = x²

If we triple the side, the side of the square each becomes 3x. 

Now, the new area, denoted by A’ is given by, 

A’ = (3x)²

A’ = 9x²

On comparing both the areas, we get, 

A’/A = 9x² / x²

= 9 

Therefore, on tripling the side, the area of the square becomes 9 times the original. 

示例问题

问题1：如果做n次边，面积增加了多少倍？

解决方案：

Since, we know,

Area of a square = (side)²

Modified area of a square, 

New area of square = (n*side)²

On comparing, we obtain, the area increases by n² times. 

问题2：根据上面的公式，如果正方形的边减半，面积会发生什么变化。

解决方案：

By straight calculation of the area of a square through its side, the area of the square is reduced and it becomes one-fourth.

This is because,

Original Area = side²

New Area = (1/2 side)²

New Area = 1/4 * Original Area

问题3：如果面积变成8倍，正方形的边受影响多少倍？

解决方案：

Let us assume the side of the square to be s.

Since, we know,

Area of a square A = s²

Now, the New area, A’ of the square is 8 times. 

Therefore, 

A’ = 8A

Let us assume the side of the new square to be s’.

s’² = 8s²

Taking square root on both sides, we have,

s’ = 2√2 s

Therefore, side of the square becomes 2√2 times. 

问题4：正方形的边被翻倍时，面积会发生什么变化？

解决方案：

Area of a square = (side)²

Let us assume the side of the square to be x and the area to be denoted by A.

On substituting the values, we get,

A = x²

If we double the side, the side of the square each becomes 2x.

Now, the new area, denoted by A’ is given by,

A’ = (2x)²

A’ = 4x²

On comparing both the areas, we get,

A’/A = 4x² / x²

= 4