Quadrilaterals and Polygons flashcards

$S=(n-2)\times 180^{\circ }$, because diagonals from one vertex split the polygon into n − 2 triangles. A hexagon has 720°, and every quadrilateral has 360°.

Exterior $=\frac{360^{\circ }}{n}$; interior $=\frac{(n-2)\times 180^{\circ }}{n}$, which equals 180° minus the exterior angle. Regular octagon: exterior 45°, interior 135°.

Subtract it from 180° to get the exterior angle, then divide 360° by that, since the exterior angles of a convex polygon total 360°. Interior 150° gives exterior 30°, so n = 12.

$D=\frac{n(n-3)}{2}$: each vertex connects to n − 3 non-neighbors, and each diagonal is counted twice. A pentagon has 5 diagonals; a decagon has 35.

Opposite sides parallel and equal, opposite angles equal, neighboring angles summing to 180°, and diagonals that bisect each other. Rectangles add equal diagonals; rhombi add perpendicular diagonals; squares have both.

$A=\frac{1}{2}(b_1+b_2)h$: the average of the two parallel bases times the perpendicular height. Bases 5 and 9 with height 4 give 7 $\times$ 4 = 28 square units.

$A=\frac{1}{2}{d}_1{d}_2$, using the full lengths of the two perpendicular diagonals. Diagonals of 12 and 8 give 48 square units.

84 square units, from A = bh with the perpendicular height. Multiplying by the slanted side (getting 108) is the classic trap; a slanted side is always longer than the height.