Defining Geometry
A subfield of mathematics called geometry examines the dimensions, placements, angles, and sizes of objects. Geometry is the area of mathematics that deals with the characteristics of space and the shapes of particular things as well as the interactions between them in space. It is one of the oldest fields of mathematics and gets its name from Greek terms that mean “Earth measuring.” And It developed in reaction to real-world issues like those in surveying. It was eventually understood that geometry need not be restricted to the study of hard three-dimensional objects and flat surfaces (plane geometry and solid geometry).
Geometrical History
Ancient peoples had already started to develop mathematical rules and techniques useful for surveying land tracts, erecting structures, and measuring storage containers, as evidenced by the earliest known unambiguous examples of written records, which date from Egypt and Mesopotamia around 3100 BCE. Beginning in the sixth century BCE, the Greeks accumulated and expanded this practical knowledge and generalized it. Geometry, which comes from the Greek terms geo (“Earth”) and metron (“measure”) for the measuring of the Earth, is now a branch of mathematics.
This article analyses certain uses of geometry in astronomy, mapping, and painting from classical Greece to medieval Islam and Renaissance Europe. In addition to outlining some of the accomplishments of the ancient Greeks, particularly Euclid’s logical formulation of geometry in the Elements. A brief discussion of contemporary extensions to non-Euclidean and multidimensional geometries finishes the article.
The physical world
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Shapes and Types
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2D Geometric Shapes
2D shapes, which include flat shapes like squares, circles, and triangles, are a subset of flat geometry. The length and width are the only 2 dimensions of these forms.
Open and closed shapes are additional categories for 2D shapes. A shape or figure that has unmet line segments and/or curves is refer to as an open shape. Their beginning and end do not coincide. Geometric shapes that start and terminate at the same location are said to be closed shapes.
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3D Geometric Shapes
A solid figure, object, or shape with three dimensions—length, breadth, and height—is refer to as a three-dimensional shape in geometry. Three-dimensional shapes contain thickness or depth, in contrast to two-dimensional shapes.
Faces, edges, and vertices are a three-dimensional figure’s characteristics. The edges of a 3D geometric object are made up of three dimensions.
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Angle
In terms of geometry, an angle is the shape created when two rays intersect at a single terminal. The sign is use to denote an angle. You can measure angles in degrees (°) with protractor. For instance, 45° is the symbol for 45 degrees.
According to their dimensions, the angles are categorized as follows:
- Less than a 90° angle is an acute angle.
- Obtuse angles range from 90° to 180°.
- A right angle is 90° in exactitude.
- A straight angle is 180° in length.
- Reflex angles range from 180° to 360°.
- A 360° angle is a full circle.
When two edges meet at a shape’s vertex, an angle results. In geometry, different shapes have different amounts of angles.
For instance: A triangle has three sides and a total interior angle of 180 degrees.
Four-sided shapes like a square, rectangles, or quadrilaterals have a total internal angle of 360 degrees.
Other polygons have different angles and different numbers of sides, such as the pentagon, hexagon, heptagon, and octagon.
In geometry, we study a variety of shape-related topics, including how to measure angles, side lengths, areas, and volumes. Geometry is characterized by similarity and congruence.
Similarity: When two shapes are the same but their sizes differ.
Congruence: When two shapes are precisely the same in terms of size and shape.
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Coordinate Plane
- A coordinate plane is a 2D surface that is created by two number lines intersecting each other at a right angle.
- The x-axis and y-axis of a number line are horizontal and vertical, respectively.
- The (0,0) location is where the two axes meet.
- We plot points, lines, etc. on the coordinate plane. We can form shapes by connecting different points on the coordinate plane.
Theorems and formulas are use to solve geometry difficulties.
A theorem is a statement that is proven using previously known facts, whereas a formula is a mathematical equation use to solve a geometrical issue.
For instance, the “Pythagoras Theorem” demonstrated that for a right-angled triangle, where a and b are its sides and c is its hypotenuse, a2+b2=c2.
The formula for determining the hypotenuse of a right-angled triangle, however, is a2+b2=c2.
Fun fact: The Greek words “Geo” and “metry,” which both imply “measuring,” are combine to get the English term “geometry.”
Solved Illustrations
Is the presented form an illustration of a straightforward closed curve that is also a polygon?
Solution:
A simple closed curve is a closed form that does not cross itself. Polygons, which include triangles, rectangles, pentagons, and other closed shapes, are only made of straight lines.
The supplied figure is not a polygon because it is curve and does not contain just straight lines.
In a triangle, ABC, right angled at B, if ∠C=45° what is the measure of ∠A?
Solution: △ ABC is the given right triangle with ∠B=90°.
Some of the angles of a triangle = 180°
∠A+∠B+∠C=180°
∠A+∠C=180°-90°=90°
But, ∠C=45° hence ∠A=90°-45°=45°
Determine the given prism’s flat surfaces.
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Solution:
The prism’s flat surfaces are listed below:
The rectangular sides of the prism are formed by rectangles AECB, DCEF, and ABDF.
△ BCD and △ AEF form the triangular faces of the prism.
Questions and Answers
What connection exists between geometry and mathematics?
The term “mathematics” refers to a broad range of academic fields with an emphasis on logic and abstract ideas. One area of mathematics that deals with the sizes, shapes, and characteristics of figures are geometry.
What are a few of the uses of geometry?
Many fields, including architecture, electronics, engineering, and construction, can benefit from the use of geometry. It can also be use to create projects or programs for space exploration in disciplines like science.
What distinguishes 2D shapes from 3D shapes in particular?
The absence of depth or height in 2D shapes is the primary distinction between them and 3D shapes. They are abstract shapes. 3D objects don’t look flat because they have three dimensions: length, breadth, and height.
What differentiates algebra from geometry?
In the field of mathematics known as algebra, variables, which take the form of letters, are use to represent numbers or quantities in equations and formulas. A subfield of mathematics called geometry examines the dimensions, placements, angles, and sizes of objects.
References
J.L. Heilbron. https://www.britannica.com/science/geometry/Relaxation-and-rigour
