哪个图形包围的面积更大：边长为 2 厘米的正方形还是长 3 厘米宽 2 厘米的长方形？

面积定义为二维形状占据的空间量。以不同的方式，我们可以说它是计算覆盖任何封闭/有界图形表面的单位正方形数量的数量。面积的标准单位是平方单位，一般用平方米、平方英尺、平方英寸、平方厘米等来表示。

闭合形状的面积是根据其长度和宽度计算的。长度是单向的，其单位是米（m）、厘米（cm）、码（yd）、英寸（in）等。但是，形状的面积是一个二维量，即长宽的乘积。因此，它以平方厘米或 (cm ² )、平方米 (m ² ) 平方英尺或 (ft ² ) 等平方单位进行测量。

如何测量任何图形的面积？

我们将在网格的帮助下理解这个概念，每个网格由许多 1 个 1 个单位的边正方形组成。每个正方形的面积是 1 个正方形单位。因此我们可以说每个方格是 1 个方格单位。看下图。让我们找出在网格中绘制的形状的面积。

彩色区域的面积是阴影单位正方形的数量

形状的面积 = 7 个平方单位。因为每个盒子是一个方格，总共有7个彩色方格，所以它的方格是7个方格。

不同几何形状的面积

如果 l 是长度，b 是宽度，h 是几何形状的高度。

广场面积

Area of square = side × side = side² square units.

For example: if side of square is 6 m then its area will be 6 × 6 m² = 36 m²

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矩形面积

Area of rectangle = length × breadth square units

for example: if the length and breadth of rectangle are 6 and 8 cm then its area will be 6 × 8 cm²= 48 cm²

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圆的面积

Area of circle = π r²square units

r is the radius of circle

for example: if radius of circle is 7 cm then its area is π 7 × 7 = 22/7 × 7 × 7 = 154 cm²

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三角形面积

Area of triangle = 1/2 × base × height = b × h square units

for example: if base and height of triangle are 4 and 6 cm respectively, then area of triangle is 1/2 × 4 × 6 = 12 cm²

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平行四边形的面积

Area of a parallelogram = base × height = b × h square units

for example if base and height of triangle are 5 and 6 cm respectively, then area of parallelogram is 5 × 6 = 30 cm².

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菱形面积

Area of Rhombus = 1/2 × (d1) × (d2) square units

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哪个图形包围的面积更大：边长为 2 厘米的正方形还是长 3 厘米宽 2 厘米的长方形？

解决方案：

First consider, square

Length of square = 2 cm

Area of square = side²

A = 2²

A = 4 cm²

Now, consider rectangle

length of rectangle = 3 cm

breadth of rectangle = 2 cm

Area of rectangle = length × breadth = 3 × 2 cm²

A’ = 6 cm²

Now after comparing values of A and A’ we see that the area of A’ is greater which means that area of rectangle is greater than square.

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示例问题

问题1：正方形的边是7厘米，计算它的面积。

解决方案：

Side of square = 7 cm

Area of a square = side × side.

A = 7 × 7 = 49

Therefore, the area of the square = 49 cm².

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问题 2：如果一个长方形的面积是 120 平方厘米，其中一个尺寸是 15 厘米，计算另一个。

解决方案：

Given,

let length of rectangle, l = 15 cm

Area of rectangle, A = 120 cm².

Area of rectangle = length × breadth

120 = 15 × b

=> b = 120/15

b = 8 cm.

Hence, other dimension of rectangle is 8cm.

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问题3：如果宽5厘米的长方形面积等于边长10厘米的正方形的面积，长方形的长度是多少？

解决方案：

Area of square = side x side

A = 10 × 10 square cm.

A = 100 square cm

Now,

Area of rectangle = length × breadth

A’ = l × 5

A’ = 5l square cm

According to question :

Area of square = Area of rectangle

=> A’ = A

5l = 100

l = 100/5 = 20 cm

Length of rectangle is 20cm

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问题 4：如果菱形的对角线是 20 厘米和 48 厘米，求菱形的周长。

解决方案：

Given diagonal d1 = 48 cm

diagonal d2 = 20 cm

Rhombus with diagonals 48 and 20 cm

OA = AD/2 = 20/2 = 10cm

OC = BC/2 = 48/2 = 24cm

In triangle AOC

Using pythagoras theorem

AC²= OA²+ OC²

AC = √(10)² + (24)²

AC = √100+ 576 = √676 = 26cm

Perimeter of rhombus = 4 × side

= 4 × 26 = 104cm

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问题5：如果正方形的边变成两倍，正方形的面积会怎样。

解决方案：

Let, the side of square be s

now the Area of square = side × side

A = s² square cm

If the side s doubles then, New side

s’ = 2s

Area of new square = side × side= s’ × s’

A’ = 2s × 2s = 4s²

A’ = 4 × A —— (A = s²)

=> Area of square will increases four times if we double its side.

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问题 6：如果正方形的边变成一半，正方形的面积会发生什么变化。

解决方案：

Let, the side of square be s

Now the Area of square = side × side

A = s² square cm

If the side s half then, New side

s’ = s/2

Area of new square = side × side = s’ × s’

A’ = s/2 × s/2 = s²/4

A’ = A/4 —— (A = s²)

=> Area of square will decrease to one-forth times if we half its side.

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