Quadrilaterals and Polygons

GRE Quantitative · Axiom Academy

LESSON Quadrilaterals and Polygons Every four-sided shape and every many-sided polygon follows the same small set of rules — learn to see them. 1. Five Quadrilaterals, One Family A quadrilateral is any closed four-sided figure. The five you need for the GRE are really just constrained versions of each other: Watch what happens when a rectangle shears sideways into a parallelogram : the base and the height never change, so the area — base × height — never changes either, even though the shape visibly tilts. Rectangle area — length × width Parallelogram area — same formula, base × height 2. Square, Rhombus & Trapezoid Area A square is the simplest case — every side is the same length s , so the area is just that side squared. A rhombus is a "squashed" square: its sides are still all equal, but its area comes from its two diagonals instead. — half the product of the diagonals. — average of the two bases, times height. The diagonals cut the rhombus into 4 right triangles — together they tile exactly half of the bounding rectangle formed by the diagonals. Rhombus with diagonals 8 and 6: area = = 24. Trapezoid with bases 5 and 7 and height 4: area = = 24. 3. Why Every Polygon's Angles Sum to Pick any vertex of a polygon and draw every diagonal from it to the other vertices. That splits the polygon into triangles — and since every triangle's angles sum to 180°, the WHOLE polygon's interior angles sum to (number of triangles) × 180° .

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