Geometry

Areas are expressed in terms of squares such as square inches, meters, miles, etc. Formulas for the areas of various plane figures are based upon the formula for the area of a rectangle, lw, where l is the length and w the width. The area of a parallelogram is bh, where b is the base and h the height (altitude), measured along a line perpendicular to the base. The area of a triangle is half that of a parallelogram with the same base and height, bh/2. When the triangle is equilateral, h = √3 b/2 so the area is √3 b²/4. A trapezoid whose parallel sides are b₁ and b₂ and whose height is h can be divided into two triangles with those bases and altitudes. Its area is (b₁ + b₂)h/2.

The area of a quadrilateral with sides a, b, c, and d depends not only on the lengths of the sides but on the size of its angles. When the quadrilateral is cyclic (all four endpoints are on in a circle), its area is given by a remarkable formula discovered by the Hindu mathematician Brahmagupta in the seventh century:

where s is the semi-perimeter (a + b +c + d)/2. This formula includes Heron's formula, discovered in the first century, for the area of a triangle,

as a special case. By letting d = 0, the quadrilateral becomes a triangle, which is always cyclic.

The area of a circle can be approximated by the area of an inscribed regular polygon. As the number of sides of this polygon increases without limit, its area approaches cr/2, where c is the circumference of the circle and r the radius. Since c = 2πr, the area of the circle is πr².

The surface area of a sphere of radius r is four times the area of a circle of the same radius, 4πr².

The lateral surface of a right circular cone can be unrolled to form a sector of a circle (see Figure 5). Its A dodecahedron is one of Plato's five regular polyhedrons (along with the tetrahedron, hexahedron, octohedron, and icosahedron): all of its faces are identically sized regular polygons. This dodecahedron has had a pentagon shaped window removed from each of its faces to allow viewing of the interior. © Richard Duncan, National Audubon Society Collection/Photo Researchers, Inc. Reproduced with permission.
area is πrs, where s is the slant height of the cone and r the radius of its base.