# Geometry

Geometry is the branch of mathematics that deals with the details of shapes. A flat surface, such as a piece of paper, is called a plane, and basic geometry is also called plane geometry.

### Cartesian coordinates

A plane is a flat, two-dimensional surface. A coordinate system is a way of describing the position of any object on a plane. The most famous and commonly-used coordinate system is Cartesian coordinates.

The system has two axis. The horizontal axis is the X axis, and the vertical axis is the Y axis. Along each axis are points, and the two axes cross each other at point (0,0). That is the origin. You get four quadrants, named I, II, III, and IV.

You can describe any position on the plane by naming coordinates. The two coordinates form an ordered pair.

### Point

The point is the starting point for all geometry. A point is an object with zero dimensions. It has no length, height, or thickness. For example, (0,0), and (5,3) are points in a coordinate system.

A point is the smallest and most precise entity in geometry.

### Line

A line is a geometric element with one dimension, length. It has no thickness and no height. In geometry, a line is conceptual, not real, so it’s infinitely long and perfectly straight.

line segment is a portion of a line. Each end of a line segment has a coordinate, such as (0,0) and (3,4). Line segments make up squares, rectangles, triangles, and other geometric shapes.

A ray is a mix of line segment and line. It has a starting point, but the other end goes to infinity.

Parallel lines are two lines that go on side by side forever and never meet. Perpendicular lines are two lines that meet at a 90-degree angle.

### Angle

An angle is a figure with two rays joined at the same endpoint. The endpoint is called the vertex. The size or amount of the angle is called the magnitude. The magnitude is the portion of a circle (360 degrees) that the angle sweeps through.

The three common angles are:

• Acute angles (less than 90 degrees) are sharp angles.
• Right angles (exactly 90 degrees) are square corners.
• Obtuse angles (more than 90 degrees but less than 180 degrees) are dull angles.

The other angle types include oblique angles, congruent angles, vertical angles, adjacent angles, complementary angles, supplementary angles, explementary (or conjugate) angles, interior angles and exterior angles.

### Square and Rectangle

A square is a rectangular quadrilateral geometric shape. It has four equal sides and four equal angles, which are always right angles. A square is a special kind of rectangle.

rectangle has two sets of sides (two equal sides for length and two equal sides for height), but unlike the square, its length and width aren’t always equal to each other. The angles are right angles.

If you write labels A, B, C, and D at the corners, you then refer to the figures as ABCD.

### Triangle

A triangle is a simple geometric shape with three sides (line segments). A triangle contains three angles, which explains the name triangle.

1. Acute triangles have three acute angles.
2. Right triangles have one 90-degree angle.
3. Obtuse triangles have one angle that’s more than 90 degrees.
4. Isosceles triangles have two equal angles.
5. Equilateral triangles have three equal (60-degree) angles.
6. Scalene triangles have three different angles.

A right triangle is part of the Pythagorean theorem, and it has special names for its parts. The two sides next to the right angle are the legs and the diagonal is called the hypotenuse.

### Polygon

A regular polygon has equal angles and equal sides. An irregular polygon has sides of different lengths and angles with different values.

A five-sided regular polygon is a pentagon. A six-sided polygon is a hexagon. An eight-sided polygon is an octagon.

### Circle

A circle is a curved geometric shape where every point on the curve is an equal distance from a fixed point called the center. The circle has the following parts:

• The circumference is the length of the line segment that constitutes the circle. It is the distance around the circle.
• The diameter is the distance across the circle through the center.
• The radius is the distance from the center to the edge of the circle.

When you know the radius you know the diameter (it’s twice as much), and when you know the diameter you know the radius (it’s half as much). When you multiply the diameter by pi (π) you know the circumference. When you divide the circumference by pi, you know the diameter.  When you know the radius, square it and multiply by pi, you know the area of the circle.

By convention, a circle has 360 degrees. A 90-degree angle sweeps one quarter of a circle, a 180-degree angle sweeps half, and a 360-degree angle is a full circle.

### Pythagorean Theorem

The Pythagorean theorem states that the sum of the squares of a right triangle’s leg lengths is equal to the square of the length of its hypotenuse.

a2 + b2 = c2

If you know the lengths of two sides of a right triangle, you can easily calculate the length of the third side.