Quadrilaterals and Polygons

Quadrilaterals

Figure	Property and area
Parallelogram	S = a · h = a · b · sin α, the diagonals bisect each other at their point of intersection
Rectangle	S = a · b, the diagonals are equal
Rhombus	all sides are equal, the diagonals are perpendicular, S = d₁ · d₂ / 2
Square	S = a², diagonal d = a√2
Trapezoid	S = (a + b) / 2 · h, midsegment (a + b) / 2

• Side of a rhombus: a² = (d₁/2)² + (d₂/2)².
• In a trapezoid, the segment joining the midpoints of the diagonals is equal to half the difference of the bases: (a − b) / 2, a > b.
• In a quadrilateral inscribed in a circle, the sum of opposite angles is 180°. In a circumscribed quadrilateral, a + c = b + d.

Example: an isosceles trapezoid with bases 13 and 7 and a leg of 5.

Quantity	Value
Bases	13 and 7
Leg	5
Half the difference of the bases	3
Height h	4

Polygons

• The sum of the interior angles of a convex n-gon is (n − 2) · 180°.
• Each interior angle of a regular n-gon is (n − 2) · 180° / n, and each exterior angle is 360° / n.
• The number of diagonals is n(n − 3) / 2.

1. Identify the figure and write down the given data
2. Find the height or the half-diagonal from a right triangle
3. Substitute into the appropriate area formula
4. Check the result using a second method

Examples

• The diagonals of a rhombus are 6 and 8: their halves are 3 and 4, the side is √(3² + 4²) = 5, the area is 6 · 8 / 2 = 24, the perimeter is 20, the height is 24 / 5 = 4.8.
• An isosceles trapezoid with bases 7 and 13 and a leg of 5: (13 − 7) / 2 = 3, h = √(25 − 9) = 4, S = 10 · 4 = 40. The midsegment is 10, and the segment joining the midpoints of the diagonals is 3.
• The angle of a regular polygon is 140°: 180n − 360 = 140n, n = 9, the number of diagonals is 9 · 6 / 2 = 27.

Watch out

• The area of a rhombus is d₁ · d₂ / 2, not a². a² applies only to a square.
• Do not forget the half-sum in the area of a trapezoid.
• The leg is not the height: the height is perpendicular to the base.
• The number of diagonals is not n(n − 1) / 2: from each vertex, no diagonal goes to the vertex itself or to its two neighbors, which is why it is n − 3.
• Do not mix up the interior and exterior angles: the exterior angle of a regular polygon is 360° / n.