GEOMETRICAL PROPERTIES OF DIFFERENT SHAPES
1. Rectangle

Area: \( A = b \times h \)
Centroid: \( \bar{x} = \frac{b}{2} \)
Centroid: \( \bar{y} = \frac{h}{2} \)
2. Circle

Area: \( A = \pi R^2 \)
Centroid: At center
3. Triangle

Area: \( A = \frac{1}{2} b h \)
Centroid: \( \bar{x} = \frac{b}{3} \)
Centroid: \( \bar{y} = \frac{h}{3} \)
4. Semi-Circle

Area: \( A = \frac{1}{2}\pi R^2 \)
Centroid: \( \bar{x} = 0 \)
Centroid: \( \bar{y} = \frac{4R}{3\pi} \)
5. Quarter-Circle

Area: \( A = \frac{1}{4} \pi R^2 \)
Centroid: \( \bar{x} = \frac{4R}{3\pi} \)
Centroid: \( \bar{y} = \frac{4R}{3\pi} \)
6. Trapezium

Area: \( A = \frac{1}{2} (a + b) h\)
Centroid: \( \bar{x} = \frac{a^2 + b^2+ab}{3(a+b)} \)
Centroid: \( \bar{y} = \frac{h(2a+b)}{3(a+b)} \)
7. Trapezium

Area: \( A = \frac{1}{2} (a + b) h\)
Centroid: \( \bar{x} = \frac{b^2 -2 a^2+ab}{3(a+b)} \)
Centroid: \( \bar{y} = \frac{h(2a+b)}{3(a+b)} \)
8. Segment

Area: \( A = R^2 (1 - \frac{\pi}{4}) \)
Centroid: \( \bar{x} = R\frac{(10-3\pi)}{12(1-\frac{\pi}{4})} \)
Centroid: \( \bar{y} = R\frac{(10-3\pi)}{12(1-\frac{\pi}{4})} \)
9. Pyramid

Volume: \( V = \frac{h}{3} (A_1+A_2+\sqrt(A_1A_2)) \)
Centroid: \( \bar{y} = \frac{h}{4}\frac{(A_1+3A_2+2\sqrt(A_1A_2))}{(A_1+A_2+\sqrt(A_1A_2))} \)