Introduction to Geometry

Anyone beginning to study geometry might be wondering what they would ever use geometry for. You need to understand geometry to get a good grade in the class, but does it have any real-world applications? Well, the architect who designed the building you’re in right now used geometry to ensure that the building was sound and wouldn’t collapse.

Basic High School Geometry

Basic geometry studies points, lines, angles, surfaces and solids.

A geometry definition of some often-used terms is shown below.

Point: A point is a specific location in space. The point is named with an upper case letter and represented by a dot, such as “point A.”

Line: A line is a series of points that continue into infinity without endpoints. Arrows at the end of a line indicate that the line extends forever. Adding two random points to the line and naming the points “A” and “F” results in line “AF.”

Line Segment: In high school geometry you will deal with many line segments. As opposed to a line that continues forever, a line segment has two endpoints. The endpoints could be named “A” and “F.”

Ray: Think of a ray of light coming from the sun. It has an endpoint (the sun) and continues forever into space away from the sun or endpoint.

Angle: An angle is simply two rays with the same endpoint, creating an angle or “v” shape.

Vertex: The vertex is the point where two rays meet.

Plane: A piece of paper that extended forever in all directions would be a plane. Arrows would indicate the plane is infinite.

Parallel Lines: Imagine a two lane highway that goes on forever. The lanes never merge or separate, but remain exactly the same distance apart. That describes parallel lines.

Intersecting Lines: An “X” is an example of two intersecting lines.

Basic Geometry Angles

Recognizing and working with different geometry angles will be important in solving many geometry problems.

Right Angle: A right angle measures 90 degrees. A 360 degree circle divided into 4 equal segments would contain 4 right angles. A carpenters square is a right angle and is used every day in construction for marking, layout and framing.

Acute Angle: An acute angle is less than 90 degrees.

Obtuse Angle: An obtuse angle refers to any angle larger than a 90 degree right angle, but less than 180 degrees.

Straight Angle: A straight angle looks like a straight line and measures 180 degrees. If a circle was cut in half, the straight side of each half-circle would be a straight angle.

Reflex Angle: A reflex angle is larger than 180 degrees, but less than 360 degrees.

Adjacent Angles: Adjacent angles share a vertex and have one side in common.

Complementary Angles: Two angles that equal 90 degrees when added together are considered complementary. The angles do not have to be adjacent.

Supplementary Angles: When added together, supplementary angles equal 180 degrees.

Vertical Angles: Vertical angles share a common vertex and use the same lines to form the sides of the angles.

Interior, Exterior and Corresponding Angles: Cross two parallel lines with a third line or transversal. You will see 8 angles.

Interior Angles: Angles 3, 4, 5 and 8

Exterior Angles: Angles 1, 2, 6 and 7

Alternate Interior Angles: Angles 3 and 5 and angles 4 and 8 are alternate interior angles. Each pair (3,5 and 4,8) is on opposite sides of the transversal – the line crossing the two parallel lines.

Alternate Exterior Angles: Angles 2 and 7 are alternate exterior angles since they are on opposite sides of the transversal.

Corresponding Angles: Angles 3 and 2 and angles 5 and 7 are corresponding angles since they hold similar positions. Polygons – Basic Geometry Shapes Take a good look around you and you’ll find polygons everywhere. Any basic geometry book will spend a lot of time on polygons. The Properties of a Polygon There are a great many polygons or geometry shapes. Polygons are considered closed plane figures. The sides of polygons can be equal or unequal in length. Regular polygon: Equal sides. Regular equiangular polygon: Equal angles. Regular equilateral polygon: Sides of the same length. Convex Polygon: If you draw a straight line through a convex polygon, you cannot cross more than 2 sides. In a convex polygon, every interior angle will be less than 180 degrees. Concave Polygon: You can draw a line through a concave polygon that will cross at least 3 sides. At least one interior angle will be greater than 180 degrees. The Parts of a Polygon Side: One of the line segments of the polygon – all polygons have at least 3 sides or line segments that don’t cross each other. Vertex: The point at which two sides meet – two or more are known as vertices. Two sides will join at every vertex. Diagonal: Any line that connects two vertices and isn’t a side. Interior Angle: The angle inside the polygon formed by two adjacent sides. Exterior Angle: The angle outside the polygon formed by two adjacent sides. The Different Types of Polygons Triangle: 3 sides – Expect to spend a lot of time working with triangles in basic geometry. There are many different types of triangles including: right, equilateral, isosceles, acute, obtuse and scalene. Quadrilateral: 4 sides Pentagon: 5 sides – The world’s most famous pentagon is the Pentagon, the headquarters building for the Dept. of Defense in Washington, DC.

Hexagon: 6 sides – A honeycomb is a hexagon with 6 equal sides and is the strongest geometrical shape on earth. Bees may have invented the hexagon, but this shape is found in everything from military tire treads to race car panels – anywhere a mechanical engineer wants to take advantage of the strength of this geometrical shape.

Heptagon: 7 sides

Octagon: 8 sides

Nonagon: 9 sides

Decagon: 10 sides

Hendecagon or 11-gon: 11 sides

Dodecagon: 12 sides

Circles

A circle is a shape with a center point and where all other points of the circle are the same distance from the center point.

Diameter: A straight line going across the circle and through the center point is the diameter of the circle.

Radius: The radius is the distance between the center point and any point on the circle. Two radius laid end-to-end will equal the diameter.

Chord: A chord is a line segment joining two points on a curve. On a circle, a chord does not pass through the center point. A chord is always shorter than the diameter of a circle.

Basic Geometry Formulas

Formulas and equations are the written language or shorthand of mathematics. Symbols are used to express a mathematical rule or relationship. When you learn any new language, it’s intimidating at first because it all looks strange and incomprehensible. The more time is spent practicing this new language, the easier and more understandable it will become.

There are countless basic geometry formulas. Fortunately, it’s not necessary to memorize each one although you will want to memorize many basic formulas. A geometry basics cheat sheet or geometry basics pdf will include those formulas that are used most often or relate to a certain geometry topic.

Equation: An equation has an equal “=” sign, meaning that the values are equal on each side of the equal sign.   2 + 3 = 5 or x + 3 = 5 are both equations. Equations can be very simple or extremely complex.

Formula: A formula is an equation that defines the relationship between differing variables. A variable is often represented by a letter such as “x” or “y” indicating that the value of the variable is not yet known. In the equation x + 3 = 5, “x” is the variable. When this equation was solved, “x” would be found to equal 2.

In geometry, formulas are used when calculating the area, volume, length or perimeter of geometric shapes and figures. A formula can be used to calculate the length of an arc, the degrees of an angle, the volume of a sphere or a polygon and for innumerable other purposes.

An equation has only one variable while a formula has at least two variables.

The subject of a formula is the single variable, usually to the left of the equal sign, which equals everything on the right side of the equal sign.

A Few Common Geometric Formulas

To calculate the volume of a box, the formula would be: v = lwh

That means: v (volume) = l (length) * w (width) * h (height)

* is the symbol for “multiply by” or “times”

A few other common geometric formulas are shown below.

Perimeter of a rectangle: l + l + w + w = 2 * l + 2 * w

Area of a rectangle: l * w

Perimeter of a square: s (side) + s + s + s = 4 * s

Area of a square: s2 or s * s

Perimeter of a parellogram: a (side “a”) + a + b (side “b”) + b = 2 * a + 2 * b

Area of a parellogram: b (base) * h (height)

Perimeter of a triangle: a + b + c (adding the lengths of the 3 sides together)

Area of a triangle: (b * h)/2 or multiplying b (base) * h (height) and then dividing that number by 2

Area of a circle: a = pir2 or a = pi*r2 or a = πr2 (these formulas are slightly different ways of saying the same thing)

Pi: pi (π) refers to the ratio of the circumference of a circle to its diameter. The numerical value of pi is a number whose digits continue forever. Pi is commonly abbreviated to 3.14159.

Becoming proficient in geometry takes practice. The basic geometry concepts build upon one another. As soon as you’re comfortable using basic geometry worksheets for one concept, you will often find yourself using what you now know how to do in the next geometry lesson. In geometry, you are solving puzzles. Accept the challenge and before you know it, you’ll be having fun and acing the course.

Geometry – Used to Solve Real-World Problems

Geometry comes from the Greek words “Earth” and “Measure.” It was conceived to solve real-world practical problems. The ancient Egyptians used early forms of geometry to build the pyramids.

Euclid wrote a geometry text “Elements” in 300 BC in which he detailed what is now called Euclidean geometry. By accepting a small set of statements or postulates as true, it’s possible to prove a great many propositions.

Geometry continued to slowly evolve, but it was almost 2,000 years before the next great advance. Rene Descartes developed coordinate geometry, which used coordinates and equations to illustrate proofs. Coordinate geometry made calculus and physics possible.

Non-Euclidean geometry was devised in the 19th century, leading to elliptical geometry and hyperbolic geometry. Elliptical or spherical geometry is used by ship captains and pilots for navigation purposes.

Jobs that Use Geometry

An understanding of basic geometry concepts will be useful in a great many jobs and real-world situations. Geometry is used in construction, architecture, geology, engineering, design, medicine, drafting, astronomy and robotics. A few of the thousands of jobs employing geometry basics are shown below.

Jewelers: Geometry is used to enable a jeweler to precisely cut the facets of a diamond or gemstone.

Fashion Designers: When designing a garment, designers create a two dimensional pattern possessing only height and width that will be cut out, stitched and fitted onto a three dimensional body that has height, width and depth. Understanding the geometry of the size and shape of clothes helps a designer to place elements such as pockets so that they create the desired effect when worn.

Designing Cars, Planes, Motorcycles and All Other Vehicles: That super-fast car or bike is the end product of a lot of math, including geometry. A computer will do most of the calculations, but the designer has to understand the principles. Formula One designs are particularly demanding since every angle and element must be exactly right in order to reach the winner’s circle.

The Military: Geometry has been a basic military skill for a very long time. Geometry is used by gunners to calculate trajectories and ranges, to build fortifications and for many other applications.

Surveyors: Surveying, whether it’s used to mark the edges of a building lot or update property lines, is all about geometry.

3D Graphic Artist, Animator or Game Developer: Geometry is used to create wire frame shapes from three dimensional real-world objects. These wire frames are then used for game characters or animations.

Geometry basics can be used for everyday tasks such as calculating how much flooring or carpet will be needed for a home renovation project. Learning and understanding high school geometry basics will come in handy many times in the future.