Geometry

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Circles

A circle is a plane curve where all points are equidistant from a point in the plane called the center. Only one circle may be drawn that passes through three noncollinear points. The word circle is sometimes used to mean the entire portion of the plane enclosed by the curve rather than just the points that lie on the curve.

Concentric circles are circles that have a common center. An angle is called a central angle of a circle if its vertex (point where the two arms of the angle meet) is at the center and its sides are radii of the circle. The circumference of a circle is divided into 360 equal degrees, and the number of degrees in a central angle is equal to the number of degrees in the intercepted arc on the circle.

The area of a circle is equal to the product of the circumference and the diameter divided by 4, or A = Cd/4. The ratio of the circumference to the diameter is approximately 3.14159265. This constant number, called pi (p), has an infinite number of nonrepeating digits. The area of a circle may also be written A = pr², where r is the radius. Similarly, the circumference is equal to the product of the diameter and the constant pi: C = pd

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Polygons

Any plane figure bounded by straight lines is a polygon. If all of a polygon’s sides are of equal length and the angles are also equal, the figure is a regular polygon. The apothem of a regular polygon is the distance from the center of the polygon to a side. The area of a regular polygon is equal to the product of one half the apothem and the perimeter, or A = ½ap:

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Triangles

A triangle is a plane figure bounded by three straight lines. A scalene triangle has three sides of unequal lengths, an isosceles triangle has two equal sides, and an equilateral triangle has three equal sides:

In the isosceles triangle the angles opposite the equal sides are equal, and in an equilateral triangle all three angles are equal.

A right triangle is a triangle in which one angle is a right angle. The side opposite the right angle is called the hypotenuse; the two adjacent sides, the legs. The famous Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs, or c² = a² + b².

The angles inside a triangle are called the interior angles; those formed by extending a side of the triangle, exterior angles. The sum of the interior angles of any triangle equals 180°. Also, an exterior angle is equal to the sum of the remote interior angles (the two interior angles that do not share a side with the exterior angle): ÐD = ÐA+ÐB.

A line drawn from a vertex of a triangle to the midpoint of the opposite side is called a median. The three medians of a triangle meet at a point two-thirds of the distance from the vertex to the midpoint of the opposite side. An altitude of a triangle is the length of the line connecting a vertex and the side opposite that vertex that is also perpendicular to the opposite side. (Two lines are perpendicular if they meet in a right angle.)

Two triangles are congruent if they satisfy any of the three following sets of conditions: (1) two angles and a side of one triangle are equal to the corresponding side and two angles of the other triangle; (2) two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle; or (3) three sides of one triangle are equal to three sides of the other triangle. If the triangles can be perfectly overlapped without removing either from their common plane, they are directly congruent; if one must be flipped over, they are inversely congruent.

If two triangles have equal angles, the triangles are said to be similar and the corresponding sides are in proportion to one another.

The area of any triangle is equal to the product of one-half of a base and the altitude perpendicular to that base: A = ybh. (Any side can be considered the base of a triangle, but usually the side on the bottom is so designated.) If the triangle is equilateral, the area is given by , where a is the length of the side. If the sides of any triangle are a, b, and c, the area is given by a relation credited to the ancient Greek mathematician Archimedes: where s is half the perimeter (s = ya + yb + yc).

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Quadrilaterals

A quadrilateral is a plane figure bounded by four straight lines. There are several familiar types of quadrilaterals. Trapezoids are quadrilaterals that have two parallel sides of unequal lengths. Parallelograms are quadrilaterals that have opposite sides of equal length. A rhombus is a parallelogram (and therefore also a quadrilateral) whose sides are equal, a rectangle is a parallelogram whose angles are all right angles, and a square is a parallelogram whose angles are right angles and whose sides are of equal length. The diagonals of a parallelogram bisect each other; if the parallelogram is a rectangle, the diagonals are also equal. Irregular quadrilaterals have four unequal and nonparallel sides:

The area of a trapezoid is half the sum of the bases times the altitude, or A = [(b₁ + b₂)/2]h. For a parallelogram, area equals base times height: A = bh.

For irregular quadrilaterals, a good method for determining the area is to divide the figure into two triangles by means of a diagonal, then find the individual areas of the triangles and add them together.

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Three-Dimensional Euclidean Figures

Figures commonly encountered in three-dimensional geometry include spheres, polyhedrons, prisms, pyramids, cylinders, and cones. Cylinders are actually special cases of prisms; cones are special cases of pyramids.