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The circle is the shape with the largest area for a given length of perimeter (see Isoperimetric inequality). The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry , and it has rotational symmetry around the centre for every angle.
A Rorschach test is a figure invariant by this symmetry, [203] as are butterfly and animal bodies more generally (at least on the surface). [204] Waves on the sea surface possess translation symmetry: moving one's viewpoint by the distance between wave crests does not change one's view of the sea. [205] Fractals possess self-similarity. [206] [207]
The method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD to find the area of a circle. [12] [13] In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method [14] [15] that would later be called Cavalieri's principle to find the volume of a sphere.
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases In this two-volume treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere and a circumscribed cylinder of the same height and diameter .
The atomic packing factor is the proportion of space filled by these spheres which can be worked out by calculating the total volume of the spheres and dividing by the volume of the cell as follows: A P F = N p a r t i c l e V p a r t i c l e V unit cell {\displaystyle \mathrm {APF} ={\frac {N_{\mathrm {particle} }V_{\mathrm {particle} }}{V ...
The multiple choice section is scored by computer, with a correct answer receiving 1 point, with omitted and incorrect answers not affecting the raw score. This total is multiplied by 1.2 to calculate the adjusted multiple-choice score. [26] The free response section is hand-graded by hundreds of AP teachers and professors each June. [27]
He regarded as his greatest achievement his finding of the surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume of a cylinder circumscribing the sphere. [69] Apollonius of Perga made significant advances in the study of conic sections.
A drainage basin is an area of land in which all flowing surface water converges to a single point, such as a river mouth, or flows into another body of water, such as a lake or ocean. A basin is separated from adjacent basins by a perimeter, the drainage divide , [ 1 ] made up of a succession of elevated features, such as ridges and hills .