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Regular polygons; Description Figure Second moment of area Comment A filled regular (equiliteral) triangle with a side length of a = = [6] The result is valid for both a horizontal and a vertical axis through the centroid, and therefore is also valid for an axis with arbitrary direction that passes through the origin.
Area rectangle area General triangular area + + [1] ... The points on the circle + = and above the axis = Arc of circle: The points on the curve (in polar coordinates ...
Rectangle; Rhomboid; Rhombus; Square (regular quadrilateral) Tangential quadrilateral; Trapezoid. Isosceles trapezoid; Trapezus; Pentagon – 5 sides; Hexagon – 6 sides Lemoine hexagon; Heptagon – 7 sides; Octagon – 8 sides; Nonagon – 9 sides; Decagon – 10 sides; Hendecagon – 11 sides; Dodecagon – 12 sides; Tridecagon – 13 sides ...
white rectangle white down ... lower half circle white square with lower left quadrant 1 ◑ ◡ ... square with upper left diagonal half black ...
An arbitrary shape. ρ is the distance to the element dA, with projections x and y on the x and y axes.. The second moment of area for an arbitrary shape R with respect to an arbitrary axis ′ (′ axis is not drawn in the adjacent image; is an axis coplanar with x and y axes and is perpendicular to the line segment) is defined as ′ = where
Squircle centred on the origin (a = b = 0) with minor radius r = 1: x 4 + y 4 = 1A squircle is a shape intermediate between a square and a circle.There are at least two definitions of "squircle" in use, one based on the superellipse, the other arising from work in optics.
The area is one half the product of the diagonals. The diagonals are perpendicular. The two line segments connecting opposite points of tangency have equal lengths. One pair of opposite tangent lengths have equal lengths. The bimedians have equal lengths. The products of opposite sides are equal.
where θ is half the sum of any two opposite angles. (The choice of which pair of opposite angles is irrelevant: if the other two angles are taken, half their sum is 180° − θ. Since cos(180° − θ) = −cos θ, we have cos 2 (180° − θ) = cos 2 θ.) This more general formula is known as Bretschneider's formula.