Trigonometry Triangles may seem like simple figures, but the mathematics behind them is deep enough to be considered its own subject: trigonometry. As the name suggests, trigonometry is the study of triangles. More specifically, trigonometry deals with the relationships between angles and sides in triangles. Somewhat surprisingly, the trigonometric ratios can also provide a richer […]
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Bhaskara II – History, Biography, and Accomplishments
Bhaskara II – History, Biography, and Accomplishments Bhaskaracharya is just one of the many historical figures that you’ll rarely hear about, unless you happen to be from his home country of India. Also known as Bhaskara II, or Bhaskara the Teacher, Bhaskaracharya formulated what would go on to become the foundations of the mathematics that […]
Girard Desargues’s Phenomenal Contributions To Geometry
Girard Desargues’s Phenomenal Contributions To Geometry Rome wasn’t built in a day, so goes the cliché, and it wouldn’t be out of place to say that mathematics and geometrics weren’t developed in a day either. Notable men of honor have helped in propagating both field of knowledge. This article is about one of the most […]
All About Yang Hui – A Nobel Chinese Mathematician
All About Yang Hui – A Nobel Chinese Mathematician Way back in math history, Yang Hui happens to be a reputable figure, one who was known for his remarkable contributions in the field of mathematics. He was a great Chinese mathematician and writer. He served through his inventions during the Song dynasty in China. So, the question is, […]
Hippocrates of Chios – History, biography and accomplishments
Hippocrates of Chios – History, biography and accomplishments Hippocrates of Chios Hippocrates of Chios was a Greek mathematician, geometer and astronomer. He grew up on the island of Chios, which is the fifth largest of the Greek islands and is much closer to Turkey than to Greece, and later moved to Athens. In Athens, he […]
The Normal Line – Definition, Applications, and Examples
This article aims to illuminate the normal line, demystifying their theoretical origins, practical applications, and the sublime beauty inherent in their geometric simplicity. Definition of the Normal Line In geometry, the normal line is perpendicular to a given line, plane, or surface at a specific point of contact. When the context involves a curve or […]
What Math Do You Take In 12th Grade – A Complete Guide
What Math Do You Take In 12th Grade? Most students’ last year of high school is the 12th, and the math courses offered to them at this point might differ from one institution to the next and from one academic track to another. Students in the 12th grade often have a variety of math courses […]
Non Adjacent Angles – Definition, Applications, and Examples
This article explores the non adjacent angles, demystifying their properties, their relationships, and their roles in shaping the geometric world as we know it. Defining Non Adjacent Angles Non adjacent angles are two or more angles that do not share a common side (ray) or a common vertex (endpoint of the rays). They are distinct […]
Cone Net – Definition, Properties, and Examples
This article invites you to explore the cone net, its properties, application, and examples. Definition of Cone Net The cone net refers to a two-dimensional pattern that can be folded along its edges to create a three-dimensional shape, in this case, a cone. The cone net generally consists of a circle (representing the base of […]
Permutation | Definition & Meaning
Permutation|Definition & Meaning Definition The permutation is a process that finds the probable arrangements in a batch when the order of the arrangements counts. Typical mathematical problems need the selection of only several objects from a batch of objects in a particular order. The permutation can be viewed as an ordered combination. The idea of […]
Geometry
Geometry Geometry is the study of points, lines, planes, and anything that can be made from those three things. Geometry includes everything from angles to trapezoids to cylinders. There is a lot of overlap with geometry and algebra because both topics include a study of lines in the coordinate plane. Geometry also provides the foundation […]
Perimeter | Definition & Meaning
Perimeter|Definition & Meaning Definition The distance along the edge of a shape that is just two dimensions is referred to as its perimeter. Additionally, it is described as the total length of all of the object’s sides when added together. The perimeter of the shape can be calculated using the algebraic addition of the lengths […]
2pir – Comprehensive Explanation and Detailed Examples
2pir is the circumference of a circle. The circumference (or the perimeter) of a circle is the total length of the circle’s boundary. The circumference is a linear measure, and its units are mostly given as centimeters, meters or inches. A circle is a closed round figure, and all the points on the circle’s boundary […]
Is -2 a Real Number? An Introduction To Real Numbers
Is -2 a real number? The answer is yes; $-2$ is a real number. Real numbers are the numbers that we use in our everyday lives. They are the numbers that we use when we count or measure things. They are the numbers that we use when we add, subtract, multiply, and divide. The Real […]
Similar Triangles Proofs – Definition and Examples
Similar triangles proof is a captivating facet of geometry, offering a rich playground for mathematical exploration and proof, embodying the harmony and beauty in the world of shapes and forms. These congruent figures, despite their varying scales, uphold the fundamental principle of similarity: maintaining identical angles while exhibiting proportional sides. Through this article, we will […]
Cos 45 Degrees – Trigonometry’s Fundamental Angle and Its Significance
by William Smith With more than 25 years of experience teaching math, I pride myself on being knowledgeable in multiple facets of this ever-growing topic. As an educator, my aim here is to share free math tutorials and lessons to assist students with their studies. The cos 45 degrees, symbolized as cos(45°) holds a special place in the fascinating world of trigonometry, we […]
Square Degrees: A Detailed Guide to This Measurement
A square degree, that is deg$^2$, is a non-SI unit of solid angle measurement. Square degrees are utilized to quantify components of a sphere in the same way that degrees are utilized to quantify components of a circle. In this complete guide, you will get to know about the degree, the square degree, and circles […]
Radians to Revolutions – Definition and Examples
In this article, we will explore radians to revolutions and delve into the relationship between radians and revolutions, their conversion factors, and their significance in various fields such as physics, engineering, and navigation. Definition of Radians to Revolutions The conversion from radians to revolutions involves determining the number of complete rotations, or revolutions, corresponding to […]
Multiply Square Roots – A Comprehensive Guide
Multiplying square roots is a fundamental concept in mathematics that involves simplifying and performing operations with expressions containing square roots. Whether you’re working on algebraic equations, geometry problems, or calculus applications, understanding how to multiply square roots is essential. The methods and guidelines for multiplying square roots will be covered in this article, along with […]
Mastering the Integration of csc(x) – A Comprehensive Guide
Welcome to an illuminating exploration of the integration of csc(x)! In the realm of calculus, the integral of the cosecant function holds intriguing properties and applications. This article delves into the world of csc(x) integration, where we will unlock its secrets and reveal the techniques required to tackle its challenges. From the fundamental concepts of […]
y = x Reflection – Definition, Process and Examples
The $boldsymbol{ y = x}$ reflection is simply “flipping” a shape or a point over a diagonal line. Since $ y= x$ reflection is a special type of reflection, it can also be classified as a rigid transformation. Knowing how to reflect over the line $y=x$ will come in handy when graphing functions and predicting […]
Cavalieri’s Principle – Definition, Conditions and Applications
The Cavalieri’s Principle relates the volumes of two solids given their cross-sections and heights. This principle is also helpful when comparing the areas of two solids given their respective bases and heights. Understanding Cavalieri’s Principle leads to a wide range of properties shared by two and three-dimensional figures. Cavalieri’s Principle states that when the two […]
Incenter Theorem – Definition, Conditions and Examples
The incenter theorem shows that the angle bisectors dividing the triangle’s vertices are concurrent. This theorem establishes the properties and formula of incenters, inradius, and even incircles. These properties and theorem open a wide range of applications and other properties of triangles. The incenter theorem states that the incenter (intersection of the triangle’s angle bisector) […]
Pythagorean Identities – Formula, Derivation, and Applications
The Pythagorean identities are important trigonometric identities that allow us to simplify trigonometric expressions, derive other trigonometric identities, and solve equations. Understanding these identities is essential when building a strong foundation to master trigonometric concepts and learn more advanced math topics. The Pythagorean identities are derived from the Pythagorean theorem. We use these identities to […]
The Hypersphere-Understanding Dimensions Beyond Three
In the awe-inspiring universe of mathematics and geometry, concepts extend beyond the standard three dimensions we experience daily. One such captivating idea is that of a hypersphere, an object existing in four or more dimensions, transcending our usual comprehension of space. Known as a higher-dimensional analogue of a sphere, the hypersphere represents a quantum leap in our understanding of geometric […]
Which Relation Is Not a Function? Explanation and Examples
In mathematics, you will come across relations and functions quite often, but one burning question that arises in many students’ minds is which relation is not a function. A relation that does not have the properties of a function is just a simple relation. Every function is a relation but every relation is not a […]
Mastering Heptagons – A Beginner’s Guide
A heptagon is a geometric figure that belongs to the broader category of polygons. Polygons are closed, two-dimensional shapes composed of straight lines. While it may seem like just another simple geometric shape, the heptagon presents fascinating complexities in its properties, historical uses, and real-world applications. Brief Overview of the Article This exploration into the […]
Exploring Quartic Function – Properties, Applications, and Examples
In the vast and interconnected realm of mathematical functions, quartic functions hold a position of unique interest and versatility. Characterized by a degree of four, these functions, defined by a fourth-degree polynomial, wield significant influence across numerous aspects of mathematical theory and its many practical applications. As the next step beyond linear, quadratic, and cubic […]
Open Interval | Definition & Meaning
Open Interval|Definition & Meaning Definition An interval that does not contain the endpoints is termed an open interval. For example, an open interval between x and y would consist of all the numbers between x and y but not including either of the numbers. Notation Open interval is generally represented with parentheses. Thus, the open […]
Cylinder | Definition & Meaning
Cylinder|Definition & Meaning Definition A cylinder is a special three-dimensional solid geometrical shape that consists of two flat circles at either end joined by a curved surface. The circles are referred to as the bases and the perpendicular distance between them represents the height of the cylinder. What Is a Cylinder? A cylinder is a […]
Width | Definition & Meaning
Width|Definition & Meaning Definition The width of a shape or object is its lateral or side-to-side distance. Since it represents length, it can be measured in any of the units of length, such as m, cm, inches, feet, etc. It can also represent the spacing between two parallel lines and other things. One way to […]
Hour Hand | Definition & Meaning
Hour Hand|Definition & Meaning Definition In an analog clock, the smallest of the three hands that refers to the hours is called an hour hand. The hour hand makes one complete revolution of the clock in 12 hours. What Is an Hour Hand? The hour hand, also known as the shorthand, is a pointer on […]
Rotational Symmetry | Definition & Meaning
Rotational Symmetry|Definition & Meaning Definition A figure which becomes identical to itself after rotation through some angel is said to have rotational symmetry. For some figures, the rotation must be through a certain angle e.g., a 90 degrees rotation for a Square produces identical figure. However, for a circle and ellipse – rotation through any […]
Reflection Symmetry | Definition & Meaning
Reflection Symmetry|Definition & Meaning Definition A geometrical figure which is identical to its reflection is said to possess reflection symmetry. For example, a square and a rectangle both have reflection symmetry since their mirror image is exactly identical to the original figures. Shapes that have reflection symmetry have a line of symmetry that divides the […]
Square Meter | Definition & Meaning
Square Meter|Definition & Meaning Definition Square meter is a unit of area denoted by m$^2$. 1 square meter is defined as the area of a square whose sides are 1 meter each. It is the most commonly used unit for the area of geometrical shapes. Why Do We Use a Square Meter? A Square meter […]
Segment | Definition & Meaning
Segment|Definition & Meaning Definition In geometry, the term segment means one of two things: part of a line (a line segment) or one of the two parts of a circle when it is split by a line or a chord of that circle (a circular segment). However, since a line segment is part of a […]
Balance Scales | Definition & Meaning
Balance Scales|Definition & Meaning Definition A device with a balanced beam and two pans hanging from it. The beam is balanced only when both pans weigh the same. Thus, with a set of reference weights, we can place an object of unknown weight in one pan and remove or add known weights in the other […]
Twice | Definition & Meaning
Twice|Definition & Meaning Definition Twice refers to the mathematical operation of multiplying a number with 2. It is also called the double or two times the number. For a number a, twice the number is written as 2a. It can also be expressed as the addition of the number with itself since 2a = a […]
Property | Definition & Meaning
Property|Definition & Meaning Definition Defining characteristics or qualities that allow us to identify a particular thing and distinguish it from others. For example, symmetry and dimensionality. A circle is perfectly symmetrical around any line passing through its center, while a trapezium has no line of symmetry. Similarly, a circle is a two-dimensional shape, whereas a […]
Metre Meter | Definition & Meaning
Metre (Meter)|Definition & Meaning Definition Meter (metre in the UK) is the standard unit of length in SI units (also called the Metric system). It is abbreviated with the letter m and officially defined as the distances covered by the light within a vacuum in exactly 1/299,792,458 parts of a second. What Is a Meter? […]
Imperial System | Definition & Meaning
Imperial System|Definition & Meaning Definition The imperial system is a system of units and measures developed in England, although it has been replaced by the metric system in almost all countries. Except for the volume measurements (all imperial volume units are larger than their US counterparts), all other units are exactly the same as in […]
Half | Definition & Meaning
Half|Definition & Meaning Definition A half is one of the two equally split parts of a whole. For example, if you halved a cake to share it with your friend, you and your friend would get half of that cake. In fractions, a half is represented as 1/2. Figure 1 – Mathematical and Graphical Representation […]
Dozen | Definition & Meaning
Dozen|Definition & Meaning Definition Exactly twelve of anything is called a dozen. This term is mostly used in the context of eggs which we usually buy “by the dozen” i.e., in packages of twelve or a dozen eggs. However, we still use it in the context of fruits and to refer to various groups of […]
Plane Shape | Definition & Meaning
Plane Shape|Definition & Meaning Definition Any two-dimensional or “flat” shape is called a plane shape. A plane has length and width but no thickness, which is true for all two-dimensional shapes as well. Therefore, we call them plane shapes. Examples include rectangles, triangles, circles, ellipses, etc. What Is a Plane Shape? In mathematics, a plane […]
Halve | Definition & Meaning
Halve|Definition & Meaning Definition To halve something means to split it into two equal parts (called halves, singular half). It is the literal opposite of double. Mathematically, it is a division by two or multiplication by the fraction 1/2 (called one-half). We work with halves every day. For example, when you share a sandwich with […]
Tessellation | Definition & Meaning
Tessellation|Definition & Meaning Definition Tessellation refers to covering a given surface with a pattern of flat shapes (either repeating or non-repeating) in such a way that no shape overlaps another, and there’s no gap between any two shapes. Real-life examples include the pattern of a brick wall (its surface is tessellated rectangles) and many types […]
Tangent Line | Definition & Meaning
Tangent Line|Definition & Meaning Definition A tangent line is one that touches any given curved line (in 2D) or curved surface (in 3D) at only one point. A tangent line has the property that its slope is equal to the slope of the curve at that tangent point. Figure 1 illustrates the difference between tangent, […]
Symmetry | Definition & Meaning
Symmetry|Definition & Meaning Definition An object has symmetry if two or more parts of it look exactly the same after flipping, sliding, or rotating one. In other words, symmetry is a property of an object that allows it to ignore the effects of manipulations and transformations and maintain a unique look. For example, your reflection […]
Subtended Angle | Definition & Meaning
Subtended Angle|Definition & Meaning Definition A subtended angle represents the angle cast by an object onto some viewpoint. Suppose a person is looking at a building. The top and bottom of the building form two lines to the person’s eyes, and the angle between these lines is the angle subtended by the building onto the […]
Surface | Definition & Meaning
Surface|Definition & Meaning Definition The outer boundary of any three-dimensional object is called the surface. It may be flat as in a pyramid or curved as in a sphere or cylinder. By definition, it is 2D and therefore has no thickness but does have area. The area occupied by the surface is generally called the […]