Geometry is a subject that students have been learning in bits and pieces since 1st grade. It is one of the most exciting branches of mathematics which children love to learn. It is also the easiest as compared to other topics of mathematics.
Geometry is one of the oldest branches of mathematics that has been taught to students for centuries. Until the 19th-century, geometry was called Euclidean Geometry that had concepts of line, point, plane, distance, angle, surface, and curve.
Later in the 19th-century, geometry saw a considerable discovery when Gauss’s theorem was introduced that students learn in their higher studies now.
What is Geometry?
Geometry deals with shapes, sizes, positions, angles, dimensions of different things. This branch of mathematics helps you to measure both 2D and 3D shapes that you see around.
The syllabus for geometry is generally divided into two halves – Plane Geometry and Solid Geometry.
What is a ray in Geometry?
In geometry, a ray can be defined as part of a line that has a beginning point but no endpoint. It can extend only in one direction. A ray can pass through more than one point.
We use rays while making angles. One angle is made out of two rays. The vertex of these angles is the starting point of these rays.
Hence, you see one angle is made of two rays. A ray is determined in words -> BA
How to draw a ray?
To draw a ray, place two points on a page and label them both with capital letters. Use a scale to draw a straight line from your starting point and continue it to your second point.
Draw an arrowhead to the open side of the line. To symbolize a ray, we identify two points on it and then write it as -> RN.
Examples of Ray
- One of the most famous examples of a ray is sunlight, which originates from the sun and goes in one direction if it is not blocked. Why do we call it sun rays and not sun lines? It is because, unlike lines, rays do not have an endpoint. The rays extend in one direction infinitely.
- Rafael Nadal, the famous tennis player, serves his tennis ball at 217 kph, which defies gravity’s tug, and hence it travels in a straight line, just like a ray.
- An LED light beam from a classroom projector is an example of a ray. A similar example is seen for a movie projector in a cinema hall.
These are some day to day examples you see around. Lasers are perfect examples to understand what rays are. They are not affected by the earth’s gravity, so they always travel in a straight line.
Lines and Segments
A line is infinite in both directions, unlike a ray. The line can be determined as <-> AB. You can mark any two points in a line and define them the same.
On the other hand, a segment has two endpoints and doesn’t extend in only one direction. You can measure the segment but not a line.
There are a few simple terms that make you understand the concept of geometry well.
- Point – a point has only one position and no particular dimension.
- Line – a line has one dimension.
- 2D Shape – a plane is two dimensional
- 3D Shape – a solid is three dimensional
There are many 2D shapes that you will find around you. Some common examples are:
Let us see some significant formulas for plane geometry.
The general formula to find out the circumference of a circle is 2πr, and for the Area, it is πr2 where “r” is the radius of the circle.
It is simple to find out the Perimeter of a triangle. You have to add all the sides together. Consider the sides as a, b, and c. Hence, the formula will be a+b+c.
When the base of a triangle and its corresponding height is given, then you can use the formula for the Area as ½ (b * h).
To find a rectangle’s perimeter, you use the formula: 2 (l+b) where l stands for length and b stands for breadth. To find the Area, we multiply, i.e., l * b
In the case of the Area of a square, you use the formula a2 where a is the side of a square. For Perimeter, you use this formula: 4a.
To find the parallelogram area, you use the formula: b * h where b stands for breadth and h for height.
To find the Area of a trapezoid, you use the formula: (1/2) (a+b) * h where a and b stands for the perpendicular opposite sides of the trapezoid, and h is the height.
Some common 3D shapes that we will learn in this article are listed below. You see them around you in daily life.
Before jumping into formulas, let us understand some basic terms of 3D figures or shapes.
- All three-dimensional figures have length, breadth, and width.
- With the help of these dimensions, you can find out its volume and surface area.
- There are two kinds of solid figures – Polyhedra (shapes with flat faces like cubes and cuboids) and non – Polyhedra (shapes that have no flat faces like spheres, cones, cylinders).
Let us see various formulas to find out the volume and surface area of these figures.
A cuboid volume is l * h * b where l stands for length h for height and b for breadth.
You can find out the cuboid’s surface area by using this formula: 2 (l * b + h * b + h * l ).
Finding a cube’s volume and surface area is the easiest because all sides of a square are the same.
For volume, you use the formula V = a3 and for surface area (SA) = 6a2 and CSA =4a2 where “a” stands for the side of a cube.
The volume of a sphere (V) = (4/3) πr3 and surface area of a sphere (SA) = 4πr2 where r stands for the radius of the sphere. The circular surface area and total surface area for a sphere are the same.
To find the volume of a right-angled cylinder, you use this formula V = πr2h and to find the surface area, use SA = 2πrh + 2πr2 = 2πr (h + r) where r stands for radius and h stands for height. The CSA is = 2πrh.
The cone volume is V = (1/3) πr2h, where r stands for radius and h stands for height. The total surface area of a cone is SA = πr (l +r) and curved surface area (CSA) = πrl.
Application of Geometry
Geometry is a vast field of mathematics. Rays are used in the most basic form in all the above 2D and 3D shapes. The examples of rays can be found easily in real life and used to determine other shapes. Its diversity is not limited to classroom studies like algebra. It is useful in real life in many ways.
- Geometry is applied in various fields like manufacturing, civil engineering, mechanical engineering, gaming, and many more.
- Different measurements and formulas are used by architects when building car parking spaces, buildings, offices, and bungalows. It helps them understand the dimensions of their designs and how much resources they will need for it.
- When a new appliance comes to your home, you check how it can fit in your home by measuring the length and breadth and width of the appliance. It is one of the common uses of geometry.
- Real Estate developers use surface area to make their customers understand how big or small the house is. They talk about a square foot of an apartment or bungalow that gives you an idea of the apartment’s size or bungalow.
- You can see geometry in almost everything around you. Geometry helps to develop medical equipment in biotechnology and medical science.
- Industries use geometry on a large scale to determine the size of factories, their products, and how much they need to sell or buy.
- In the manufacturing industry, the volume of contents is measured in tons, metric tons, and orders are placed accordingly; without geometry, such massive data is tough to produce.
Hence, it plays an important role in daily life and the primary and secondary sector of the economy in any country.
You can learn more about Geometry Formulas and Examples from basics to advance and get an interesting insight to understand geometry concepts quickly.