Euclid's Geometry
Euclid's Geometry, also known as Euclidean Geometry, is considered the study of plane and solid shapes based on different axioms and theorems. The word Geometry comes from the Greek words 'geo’, meaning the ‘earth’, and ‘metrein’, meaning ‘to measure’. Euclid's Geometry was introduced by the Greek mathematician Euclid, where Euclid defined a basic set of rules and theorems for a proper study of geometry. In this section, we are going to learn more about the concept of Euclid's Geometry, the axioms and solve a few examples.
| 1. | What is Euclid's Geometry? |
| 2. | Definition of Euclid's Geometry |
| 3. | Euclid's Axioms |
| 4. | Euclid's Postulate |
| 5. | Non-Euclidean Geometry |
| 6. | FAQs on Euclid's Geometry |
What is Euclid's Geometry?
Euclid's Geometry was introduced by the Father of Geometry i.e. Euclid and is also called Euclidean Geometry. Geometry was originated from the need for measuring land and was studied in various forms in every ancient civilization such as Egypt, Babylonia, India, etc. Euclid's geometry came into play when Euclid accumulated all the concepts and fundamentals of geometry into a book called 'Elements'. This book spoke about the definitions, the axioms, the theorems, and the proof of various shapes. Euclid specifically spoke about the shape, size, and position of solid shapes and various terms associated with them such as the surface, straight or curved lines, points, etc. Some of his fundamentals about solid shapes are :
- A point has no parts.
- A line is a breadthless length.
- The ends of a line are points.
- A straight line is a line that lies evenly with the points on itself.
- A surface has a length and breadth only.
- The edges of a surface are lines.
- A plane surface is a surface that lies evenly with the straight lines on itself.
Definition of Euclid's Geometry
Euclid's geometry or the euclidean geometry is the study of Geometry based on the undefined terms such as points, lines, and planes of flat spaces. In other words, it is the study of geometrical shapes both plane shapes and solid shapes and the relationship between these shapes in terms of lines, points, and surfaces. Euclid introduced axioms and postulates for these solid shapes in his book elements that help in defining geometric shapes. Euclid's geometry deals with two main aspects - plane geometry and solid geometry. The table below mentions the theorems that were proved by Euclid.
| Plane Geometry | Theorem Proved |
| Congruence of Triangles | Two triangles are congruent if they are similar in shape and size. |
| Similarity of Triangles | Two triangles are similar in shape but differ in size. |
| Areas | Area of a plane shape can be measured by comparing it with a unit square. |
| Pythagorean Theorem | Pythagorean theorem helps in calculating the distance in different situations for Geometric shapes. |
| Circles | Equal chord determines equal angles and vice versa in a circle. |
| Regular Polygons | Regular Polygons are equal in sides and angles. |
| Conic Section | Conic sections include Ellipse, Parabola, and Hyperbola. |
| Solid Geometry | Theorem Proved |
| Volume | Volume of a shape can be calculated. |
| Regular Solids | The existence of Platonic Solids. |
Euclid's Axioms
Euclid's axioms or common notions are the assumptions of the obvious universal truths that have not been proven. But in his book, Elements, Euclid wrote a few axioms or common notions related to geometric shapes. Let us take a look:
Axiom 1: Things that are equal to the same thing are equal to one another.
Suppose the area of a rectangle is equal to the area of a triangle and the area of that triangle is equal to the area of a square. After applying the first axiom, we can say that that the area of the triangle and the square are equal. For example, if p = q and q = r, then we can say p = r.
Axiom 2: If equals are added to equals, the wholes are equal.
Let us look at the line segment AB, where AP = QB. When PQ is added to both sides, then according to axiom 2, AP + PQ = QB + PQ i.e AQ = PB.
Axiom 3: If equals are subtracted from equals, the remainders are equal.
Consider rectangles ABCD and PQRS, where the areas are equal. If the triangle XYZ is removed from both the rectangles then according to axiom 3, the areas of the remaining portions of the two triangles are equal.
Axiom 4: Things that coincide with one another are equal to one another.
Consider line segment AB with C in the center. AC + CB coincides with the line segment AB. Thus by axiom 4, we can say that AC + CB = AB.
Axiom 5: The whole is greater than the part.
Using the same figure as above, AC is a part of AB. Thus according to axiom 5, we can say that AB > AC.
Axiom 6 and Axiom 7: Things that are double of the same things are equal to one another. Things that are halves of the same things are equal to one another.
Axiom 6 and 7 are interrelated. Consider two identical circles with radii \((r)_{1}\) and \((r)_{2}\) with diameters as \((d)_{1}\) and \((d)_{2}\) respectively. Since the circles are identical, using both axioms 6 and 7, we can say that
\((r)_{1}\) = \((r)_{2}\) and \((d)_{1}\) = \((d)_{2}\).
Euclid's Postulate
For discussing Euclid's postulate, there are a few terms that we need to get familiarized with. Euclid talks about a three-step process from solids to points which is solids-surface-lines-points. At each step, one dimension is lost from the shape. Therefore, a solid is a 3D shape, a surface is a 2D shape, a line is a one dimension shape, and points are dimensions. The term surface means something that has length and breadth only. Whereas a point has no part, has a long length, etc. These terms will help in understanding the postulate better. There is 5 Euclid's postulate, let us take a look:
Postulate 1: A straight line segment can be drawn for any two given points.
This postulate shows us that at least one straight line passes through two distinct points, but it does not say that there cannot be more than one such line. Look at the line below, only one line passes through P and Q which is PQ that passes through both Q and P respectively.
Postulate 2: A line segment can be extended in either direction to form a line.
A line segment can be extended in either direction to form a line is the second postulate.
Postulate 3: To describe a circle with any center and radius.
A circle is considered as a plane figure that consists of a set of points that are equidistant from a reference point and can be drawn with its center and radius. According to the third postulate, the shape of a circle does not change when the radius is different. What changes is the size of the circle.
Postulate 4: All right angles are equal to one another.
A right-angle measures at exactly 90° irrespective of the lengths of their arms. Hence according to postulate 4, all right angles are equal to each other. This holds good only for right-angled triangles and not acute angle triangles or obtuse angle triangles.
Postulate 5: If two lines are intersected by a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines will intersect each other on that side if produced indefinitely.
When there are two lines cut by a third line, if the sum of the interior angles is less than 180°, then the two lines will meet when extended on that side.
In the image given below, \(\angle 1 + \angle 2 < 180^{\circ}\). Therefore, Line \(m\) and \(n\) will meet when extended on the side of 1 and 2
Non-Euclidean Geometry
There is a branch of geometry known as Non-Euclidean geometry. Basically, it is everything that does not fall under Euclidean geometry. However, it is commonly used to describe spherical geometry and hyperbolic geometry. Since spherical geometry comes under non-euclidean geometry, to convert it to euclidean or Euclid's geometry or basic geometry we need to change actual distances, location of points, area of the regions, and actual angles.
Related Topics
Listed below are a few interesting topics related to Euclid's geometry, have a look.
Examples on Euclid's Geometry
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Example 1: Bella marked three points A, B, and C on a line such that, B lies between A and C. Help Bella prove that AB + BC = AC.
Solution
AC coincides with AB + BC.
Euclid’s Axiom (4) says that things that coincide with one another are equal to one another. So, it can be deduced that AB + BC = AC
It has been assumed that there is a unique line passing through two points.
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Example 2: Prove that an equilateral triangle can be constructed on any given line segment.
Solution:
A line segment of any length is given, say AB. Using Euclid’s postulate 3, first, draw an arc with point A as the center and AB as the radius. Similarly, draw another arc with point B as the center and BA as the radius. Mark the meeting point of the arcs as C. Now, draw the line segments AC and BC to form \(\triangle \text{ABC}\).
AB = AC; Arcs of same length. AB = BC; Arcs of same length.
Euclid’s axiom says that things which are equal to the same things are equal to one another. Hence, AB = BC = AC. Therefore, \(\triangle \text{ABC}\) is an equilateral triangle.
FAQs on Euclid's Geometry
What is Euclid's Geometry?
Euclid's geometry is a type of geometry started by Greek mathematician Euclid. It is the study of planes and solid figures on the basis of axioms and postulates invited by Euclid. Euclid's geometry is also called Euclidean Geometry. He defined a basic set of rules and theorems for a proper study of geometry through his axioms and postulates.
What are the 7 Axioms of Euclids?
Axioms or common notions are theories made by Euclid that may or may not be used in geometry. The 7 axioms are:
Things that are equal to the same thing are equal to one another.
- If equals are added to equals, the wholes are equal.
- If equals are subtracted from equals, the remainders are equal.
- Things that coincide with one another are equal to one another.
- The whole is greater than the part.
- Things that are double of the same things are equal to one another.
- Things that are halves of the same things are equal to one another.
What are the 5 Postulates of Euclid's Geometry?
Euclid's 5 postulates are:
- A straight line segment can be drawn for any two given points.
- A line segment can be extended in either direction to form a line.
- To describe a circle with any center and radius.
- All right angles are equal to one another.
- If two lines are intersected by a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines will intersect each other on that side if produced indefinitely.
What is Meant by Non-Euclidean Geometry?
Non-euclidean geometry is another branch of geometry that is completely opposite to Euclid's geometry. This branch of geometry talks about spherical geometry and hyperbolic geometry.
What are the Fundamentals of Solid Shapes in Euclid's Geometry?
Some of the fundamentals of solid shapes according to Euclid's geometry are:
- A point has no parts.
- A line is a breadthless length.
- The ends of a line are points.
- A straight line is a line that lies evenly with the points on itself.
- A surface has a length and breadth only.
- The edges of a surface are lines.
- A plane surface is a surface that lies evenly with the straight lines on itself.
What Geometry Did Euclid Do?
Euclid worked on theorems to create Euclid's Geometry which is the basic form of geometry that deals with planes and solid figures. Euclid worked on different axioms or common notions and postulates or theorems to make geometry simple and easy to use.