Geometry

Geometry

This branch of mathematics deals with points, lines, planes, and figures and their properties, measurement, and spatial relationships.

Angles: Angles are expressed in degrees, which are fractions of a circle. A circle has 360 degrees.

An acute angle is greater than zero degrees and less than 90.

A right angle has 90 degrees. The lines forming the angle run perpendicular to each other.

An obtuse angle has more than 90 degrees but less than 180 degrees.

A straight angle has 180 degrees and forms a straight line.

Complementary angles exist when two angles total 90 degrees.

Supplementary angles occur when two angles add up to 180 degrees.

Conjugate angles add up to 360 degrees when combined.

Triangles: The sum of the internal angles of a triangle is always 180 degrees.

An equilateral triangle has sides that are of equal length and internal angles that are all 60 degrees.

An isosceles triangle has two side that are of equal length and two equal angles.

A scalene triangle has no sides and no angles of equal size.

A right triangle has one internal angle of 90 degrees.

An obtuse triangle has one obtuse angle, which is an angle greater than 90 degrees but less than 180 degrees.

An acute-angle triangle has three acute angles, meaning that all are under 90 degrees.

To calculate the area of a triangle, multiply the base by the height by one-half.

Geometrical figures. Artwork by Elena

Quadrilaterals: A quadrilateral is a four-sided polygon.

A square has four equal sides and four right angles. To calculate the area of a square, square the length of one side.

A rectangle has equal opposite sides and all right sides. To calculate a rectangle`s area, multiply base by height.

A rhombus has equal sides and no right angles. To calculate its area, multiply base by height.

A parallelogram has opposite sides that are parallel to each other and are the same length. To calculate the area of a parallelogram, multiply the base by the height.

Other polygons

A pentagon is a five-sided polygon. To calculate the approximate area of a pentagon, multiply the square of the length of one side by 1.721,

A hexagon is a six-sided polygon. To calculate the approximate area of a hexagon, multiply the square of the length of one side by 2.598.

An octagon is an eight-sided polygon. To calculate the approximate area of an octagon, multiply the square of the length of one side by 4.828.

A circle is a figure in which every point units boundary is equidistant from the center. The radius is that distance to the center. The diameter is twice the radius, or the longest distance across the circle. The circumference is the total distance around the boundary of the circle. To calculate the area of a circle, multiply the square of the radius by pi (3.1416…). To calculate the circumference of a circle, multiply radius of the circle py 2 and by pi (3.1416…).

The Ins and Outs of a Right Triangle

The relationships between the angles of a right triangle and its sides have been studied by mathematicians for millennia. Among the important trigonometric concepts are:

Sine: Ina right triangle, the ratio of the opposite side of a given acute angle to the hypotenuse is known as the sine of that angle.

Cosine: In a right triangle, the ration of the adjacent side of an acute angle to the hypotenuse is known as the cosine of that angle.

Tangent: In a right triangle, the ration of the opposite to the adjacent side of an acute angle is known as the angle’s tangent.

Cotangent: In a right triangle, the ration of the adjacent side of an acute angle to the opposite side is known as the angle’s cotangent.

The Pythagorean Theorem: In a right triangle the square of the hypotenuse is equal to the sum of the square of the other two sides.

Solids

Solids are three-dimensional geometric objects that exist in space.

A cube is a solid with six equal, square sides. To calculate the surface area of a cube, multiply the square of the length of one side by 6.

To calculate the volume of a cube, cube the length of one side.

A sphere is a body whose surface is equally distant from the center at all points. To calculate the surface area of a sphere, multiply 4 by pi (3.1416) by the square of the radius. To calculate the volume of a sphere, multiply the cube of the radius by pi by 4/3.

A pyramid is a shape with a square base and four sloping triangular sides meeting at the top. To calculate the surface area of a pyramid, multiply base by length, and multiply 2 by the base and by the height, and add the results. To calculate a pyramid’s volume, multiply the base’s area by height and by 1/3.

A cylinder is a solid described by a line that always has a point in common with a given closed curve. To calculate the surface area of a right, circular cylinder, multiply 2 by pi by radius by height. To calculate the volume, multiply the square of the radius of the base by pi by height. To calculate the volume, multiply the square of the radius of the base by pi by height.

A cone is a flat-based, single-pointed solid formed by a rotating straight line that traces out a base from a fixed vertex point. To calculate the surface area of a cone, multiply pi by the radius of the base by the slant height. To calculate the volume of a cone, multiply the square of the radius of the base by pi by height by 1/3.