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PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b. Using the geometric mean theorem, triangle PGR's altitude GQ is the geometric mean. For any ratio a:b, AO ≥ GQ. A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-edge and compass.
Area#Area formulas – Size of a two-dimensional surface; Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities
The arc length, from the familiar geometry of a circle, is s = θ R {\displaystyle s={\theta }R} The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of θ {\displaystyle \theta } ):
where C is the circumference of an ellipse with semi-major axis a and semi-minor axis b and , are the arithmetic and geometric iterations of (,), the arithmetic-geometric mean of a and b with the initial values = and =.
In analogy with the cross-section of a solid, the cross-section of an n-dimensional body in an n-dimensional space is the non-empty intersection of the body with a hyperplane (an (n − 1)-dimensional subspace). This concept has sometimes been used to help visualize aspects of higher dimensional spaces. [7]
Section Sub-Section Sup-Section Name Algebraic Curves ¿ Curves ¿ Curves: Cubic Plane Curve: Quartic Plane Curve: Rational Curves: Degree 2: Conic Section(s) Unit Circle: Unit Hyperbola: Degree 3: Folium of Descartes: Cissoid of Diocles: Conchoid of de Sluze: Right Strophoid: Semicubical Parabola: Serpentine Curve: Trident Curve: Trisectrix of ...
The following is a list of centroids of various two-dimensional and three-dimensional objects. The centroid of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane.
A standard example is the Reuleaux triangle, the intersection of three circles, each centered where the other two circles cross. [2] Its boundary curve consists of three arcs of these circles, meeting at 120° angles, so it is not smooth, and in fact these angles are the sharpest possible for any curve of constant width. [3]