Quadrilaterals & Areas

At a glance 11 formulas

1. 1ParallelogramOpposite sides are equal and parallel, opposite angles are equal, neighbouring angles add to $180^\circ$, and the diagonals bisect each other.
2. 2Parallelogram area$\text{base} \times \text{height} = ab\sin\theta$$a, b = \text{sides},\ \theta = \text{angle between them}$
3. 3Parallelogram diagonals$d_1^2 + d_2^2 = 2(a^2 + b^2)$$d_1, d_2 = \text{diagonals},\ a, b = \text{sides}$
4. 4Rhombus and square$\text{area} = \tfrac{1}{2} d_1 d_2\ (d_1, d_2 \text{ diagonals})$$\text{side}^2 = \tfrac{d_1^2 + d_2^2}{4};\ \text{square, diagonal } d:\ \tfrac{d^2}{2}$
5. 5Trapezium$\text{area} = \tfrac{1}{2}(a + b)h,\ h = \text{height}$$a, b = \text{parallel sides};\ \text{diagonals cut } a : b$
6. 6Any quadrilateral$\text{area} = \tfrac{1}{2} d_1 d_2 \sin\theta$$d_1, d_2 = \text{diagonals},\ \theta \text{ between them}$
7. 7Triangles at the diagonals$AOB \times COD = BOC \times DOA$$\text{least total} = \left(\sqrt{AOB} + \sqrt{COD}\right)^2$
8. 8Between the same parallelsA triangle on the same base as a parallelogram, with its third corner on the opposite side, has half the parallelogram's area.
9. 9Midpoint quadrilateralJoining the midpoints of any quadrilateral's sides in order gives a parallelogram with sides half the diagonals and half the area.
10. 10Square in a square$\text{inner area} = a^2 + b^2$$a, b = \text{the two parts of each side}$
11. 11Trapezium, perpendicular diagonals$\text{area} = h^2 = \left(\tfrac{a + b}{2}\right)^2$$\text{isosceles: } a, b = \text{parallel sides},\ h = \text{height}$