梯形公式的质心

梯形是一种具有两条平行边的四边形。四边形是一种具有四个边的多边形。四边形的四个内角之和是360°。质心是指任何图形的中心点。它也被称为几何中心。在 2D 坐标系中，质心始终相对于 x 轴和 y 轴进行测量。在梯形中，质心也相对于这两个轴进行测量。

梯形公式的质心

质心点是梯形的中心点。质心以梯形中心点坐标的形式表示。通过质心点的水平或垂直线将其分成两个相等的部分。这意味着，对于高度“h”，中心点将是“h/2”。类似地，计算另一个中心点。这两个点形成质心。

梯形图如下所示：

图形的质心由下式给出：

C = [X_C, Y_C]

Where,

X_C is the point on the x-axis

Y_C is the point on the y-axis

The formula to calculate the centroid of a trapezoid is given by:

C = [h/2, $\frac{(q + 2p)h}{3(p + q)}$]

Where,

h is the height

p is the base  

q is the opposite parallel side  

Comparing the values, we get:

X_C = h/2

Y_C = $\frac{(q + 2p)h}{3(p + q)}$

示例问题

问题1：求一个梯形的质心，高4m，两条平行边分别为5m和3m。

解决方案：

The formula to calculate the centroid of a trapezoid is given by:

C = [h/2, $\frac{(q + 2p)h}{3(p + q)}$]

Where,

h = height

p and q are the two parallel sides  

h = 4m

p = 5m

q = 3m

C = [4/2,$\frac{(3+ 2\times5)4}{3(5 + 3)}$]

C = [2, 52/24]

C = [2, 2.17]

Thus, the centroid is [2, 2.17] or 2m with respect to the x-axis and 2.17m with respect to the y-axis.  

问题2：求一个梯形的质心，高2cm，两条平行边分别为6cm和4cm。

解决方案：

The formula to calculate the centroid of a trapezoid is given by:

C =  [h/2,$\frac{(q + 2p)h}{3(p + q)}$]

Where,

h = height

p and q are the two parallel sides  

h = 2cm

p = 6cm

q = 4cm

C = [2/2, $\frac{(4 + 2 \times 6)2}{3(6 + 4)}$]

C = [1, 32/30]

C = [1, 1.07]

Thus, the centroid is [1, 1.07] or 1cm with respect to the x-axis and 1.07cm with respect to the y-axis.  

问题3：求一个梯形的质心，高10m，两条平行边分别为7m和4m。

解决方案：

The formula to calculate the centroid of a trapezoid is given by:

C = [h/2, $\frac{(q + 2p)h}{3(p + q)}$]

Where,

h = height

p and q are the two parallel sides  

h = 10m

p = 7m

q = 4m

C = [10/2, $\frac{(4 + 2 \times 7)10}{3(7 + 4)}$]

C = [5, 180/33]

C = [5, 5.45]

Thus, the centroid is [5, 5.45] or 5m with respect to the x-axis and 5.45m with respect to the y-axis.  

问题4：求一个梯形的质心，高11cm，两条平行边分别为3cm和2cm。

解决方案：

The formula to calculate the centroid of a trapezoid is given by:

C =  [h/2,$\frac{(q + 2p)h}{3(p + q)}$]

Where,

h = height

p and q are the two parallel sides  

h = 1cm

p = 2cm

q = 3cm

C = [1/2, $\frac{(3 + 2 \times 2)1}{3(2 + 3)}$]

C = [0.5, 7/15]

C = [0.5, 0.47]

Thus, the centroid is [0.5, 0.47] or 0.5cm with respect to x-axis and 0.47cm with respect to y-axis.  

问题 5：求一个梯形的质心，高 8m，两条平行边分别为 5m 和 3m。

解决方案：

The formula to calculate the centroid of a trapezoid is given by:

C =  [h/2,$\frac{(q + 2p)h}{3(p + q)}$]

Where,

h = height

p and q are the two parallel sides  

h = 8m

p = 5m

q = 3m

C = [8/2, $\frac{(3 + 2 \times 5)8}{3(5 + 3)}$]

C = [4, 104/24]

C = [4, 4.33]

Thus, the centroid is [4, 4.33] or 4m with respect to the x-axis and 4.33m with respect to the y-axis.  

问题 6：求一个梯形的质心，梯形高 5m，两条平行边分别为 7m 和 9m。

解决方案：

The formula to calculate the centroid of a trapezoid is given by:

C = [h/2, $\frac{(q + 2p)h}{3(p + q)}$]

Where,

h = height

p and q are the two parallel sides  

h = 5m

p = 7m

q = 9m

C = [5/2, $\frac{(9 + 2 \times 7)5}{3(7 + 9)}$]

C = [2.5, 115/48]

C = [2.5, 2.4]

Thus, the centroid is [2.5, 2.4] or 2.5m with respect to the x-axis and 2.4m with respect to the y-axis. 

问题7：求一个梯形的质心，高20cm，两条平行边分别为15cm和12cm。

解决方案：

The formula to calculate the centroid of a trapezoid is given by:

C =  [h/2, $\frac{(q + 2p)h}{3(p + q)}$]

Where,

h = height

p and q are the two parallel sides  

h = 20cm

p = 15cm

q = 12cm

C = [20/2, $\frac{(12 + 2 \times 15)20}{3(15 + 12)}$]

C = [10, 840/81]

C = [10, 10.37]

Thus, the centroid is [10, 10.37] or 10cm with respect to the x-axis and 10.37cm with respect to the y-axis.