Search results for: circle

Surd|Definition & Meaning Definition A surd is any number whose nth root (square root, cube root, etc.) cannot be simplified so as to remove the radical sign, e.g., $sqrt{2}$ and $sqrt[3]{4}$. Unless the radicand can be factorized into parts that are all perfectly expressible as a power of the radical’s index (perfect squares for square […]

Square Measure|Definition & Meaning Definition Any unit that measures area is called a square measure. Therefore, all squared units of length (e.g., squared meters or m$^2$, squared centimeters or cm$^2$) and some special units like hectares (ha = 100 m x 100 m = 10,000 square meters) and acres (ac = 0.4047 hectares) are called […]

Solid|Definition & Meaning Definition Any three-dimensional object is generally called a solid. These are shapes as we perceive them around us. Unlike 2D shapes, solids have the third dimension of depth in addition to length and width, which leads to the concept of volume. Pyramids, cylinders, cubes, spheres, etc., are all examples of solid 3D […]

Sinh|Definition & Meaning Definition Sinh (pronounced as “shine”) represents the hyperbolic sine function. While the trigonometric function sine is defined as a circle, sinh is related to measurements from a hyperbola. It involves the natural exponential term e$^x$: [ sinh(x) = frac{e^x-e^{-x}}{2} ] What Is a Hyperbolic Function? Similar to the circle and trigonometric functions, […]

Similar|Definition & Meaning Definition In geometry, two shapes or objects are similar if re-scaling, flipping, translating, or rotating one of them results in the other shape or object. For example, the corresponding sides of similar triangles are proportional (i.e., corresponding angles are equal). This implies that we can form infinitely many shapes from just one […]

Semi|Definition & Meaning Definition Semi literally means “half.” It is used as a prefix to imply halves of various things. For example, semi-annually means every half-year (e.g., interest compounding), semicircle means a half circle, and semi-major and minor axes mean half the major and minor axes of ellipses. It is also used in other contexts, […]

Sector|Definition & Meaning Definition A sector of a circle is a closed region bounded by an arc and any two radii connecting the ends of that arc to the center of the circle. It looks like a slice of cake or pizza. The area of a sector is always less than the circle’s area. The […]

Row|Definition & Meaning Definition In mathematics, a row is a general term describing things that lie horizontally adjacent (left-right) to each other. This is contrary to columns, which describe things lying vertically adjacent (up-down) to each other. A row refers to the way that items can be arranged. Although rows and columns are frequently combined […]

Rhombus|Definition & Meaning Definition A rhombus is a two-dimensional (or flat), closed shape with four equal-length straight sides such that the opposite angles are equal and the opposite sides are parallel to each other. In other words, it is a parallelogram with all four sides equal. A square is a type of rhombus where all […]

Revolution|Definition & Meaning Definition A revolution means a full circular rotation such that you end up in the same position you started turning from. Usually, the object is spinning around some axis. In two dimensions, a revolution is a full 360-degree rotation about some center point. For example, if you rotate a point on a […]

Parametric Equation – Explanation and Examples In mathematics, a parametric equation is explained as:  “A form of the equation that has an independent variable in terms of which any other equation is defined, and dependent variables involved in such an equation are continuous functions of the independent parameter.” For instance, let us consider the equation […]

Polar Coordinates – Definition, Conversion, and Examples Polar coordinates help us represent objects or relationships that are centrosymmetric (symmetric with respect to a common center).  The polar coordinate system, along with the rectangular coordinate system, is one of the most used and most helpful coordinate systems there are. In fact, there are instances when a […]

Polar Curves – Definition, Types of Polar Curves, and Examples Polar curves give us a better understanding of how we graph equations with a different coordinate system than what we’re used to. In the past, we’ve learned about polar coordinates, so it’s time for us to expand our graphing skills on the polar coordinate system. […]

Quadric Surfaces – Definition, Types, and Examples Knowing about common quadric surfaces will help us understand how second-degree equations can be graphed in space. Ellipsoids and hyperboloids are just two of the many quadric surfaces that are widely used in early computer-aided architecture and engineering models. Of course, learning about quadric surfaces will also help […]

Parametric Curves – Definition, Graphs, and Examples Learning about parametric curves will give us one more with special attributes (time to be specific). There are instances when modeling quantities using parametric curves is more helpful than graphing them in the coordinate systems that we know – rectangular and polar coordinate systems. This is why it’s […]

Vector Function – Definition, Properties, and Explanation The vector functions allow us to visualize functions in two or three-dimensional coordinate systems and account for another element: the direction of the curve. Similar to parametric functions, vector functions depend on parameters to visualize their graphs.  These functions are most helpful when we’re observing different types of […]

Lagrange Multipliers – Definition, Optimization Problems, and Examples The method of Lagrange multipliers allows us to address optimization problems in different fields of applications. This includes physics, economics, and information theory. Seeing the wide range of applications this method opens up for us, it’s important that we understand the process of finding extreme values using […]

Triple Integral – Definition, General Forms, and Examples Through triple integrals, we can now extend our understanding of iterated integrals to expressions and functions with three variables. We can use triple integrals to calculate the volume and mass of different objects – which are important components we need in physics, structural engineering, and other fields. […]

Change of Variables in Multiple Integrals – Technique, Steps, and Examples Knowing how to change variables in multiple integrals allows us to simplify our process of integrating complex functions. There are instances when we need to rewrite the integral of a function in the Cartesian form to its polar form so that we can easily […]

Vector Fields – Definition, Graphing Technique, and Example Vector fields allow us to exhibit relationships between objects that expand over a wide region of the plane (or even space). In Physics, for example, we use vector fields to describe the object’s magnetic or electric fields. These are also essential when studying large-scale occurrences such as […]

Line Integral – Definition, Properties, and Examples Learning about the line integral opens a wide range of functions that we can now study. In physics and engineering, we use line integrals to estimate the work of a particle in a force field or a mass in a gravitational field. Aside from its extensive applications, we […]

Fundamental Theorem for Line Integrals – Theorem and Examples The fundamental theorem of line integrals shows us how we can extend the fundamental theorem of calculus when evaluating line integrals.  By generalizing the fundamental theorem of calculus to include line integrals, we can also establish interesting properties about a line integral’s paths. Line integrals are […]

Green’s theorem – Theorem, Applications, and Examples Green’s Theorem is one of the most important theorems that you’ll learn in vector calculus.  This theorem helps us understand how line and surface integrals relate to each other. When a line integral is challenging to evaluate, Green’s theorem allows us to rewrite to a form that is […]

Center of Mass – Definition, Formula, and Examples Integral calculus has a wide range of applications in physics and engineering and that includes finding the center of mass through integration. In this article, we’ll learn how to compute the center of mass of a given object through single and double integration. The center of mass […]

Venn diagram – Explanation & Examples The definition of the Venn diagram is: “Venn diagram is an illustration that shows the relation between two or more groups using overlapping circles.” In this topic, we will discuss the Venn diagram from the following aspects: What is a Venn diagram? Venn diagram components How to read a Venn […]

Universal Set – Definition and Examples In previous lessons, we learned that a set is a collection of objects and that it is possible to use Venn diagrams to explain both set relationships and logical relationships. We also learned about various types of sets. In Mathematics, all these different types of sets have their value. […]

Intersection Of Sets – Definition and Examples The concept of intersection is not only limited to set theory. It is, in fact, a widespread mathematical concept. It is one of the prevalent concepts of Euclidean geometry and set theory. Intuitively speaking, whenever we intersect two or more objects, it usually results in a smaller object.  […]

Symmetric Difference – Definition and Examples If you have been keeping up with the lessons, we’re confident you’re gradually progressing in your understanding. So far, we have covered several definitions, including the union of two sets, intersection, complement, etc. We have learned the difference between disjoint sets and intersecting sets. We have also discussed, at […]

Union of Sets – Definition and Examples We looked at sets before, and they can be defined as the collection of distinct and unique elements. These elements can be numbers, alphabets, addresses of city halls, locations of stars in the sky, or numbers of electrons in a certain atom. We also discussed that we could […]

Union Vs Intersection – Explanation and Examples In sets and set theory, two of the most fundamental concepts are set union and set intersection. These two concepts build the foundation of sets and give birth to the pictorial representation of sets in Venn diagrams. The battle between union and intersection is a prolonged one that […]

Vector Equations – Explanation and Examples In vector geometry, one of the most important concepts in solving real-world problems is using vector equations. The vector equation is defined as: “The vector equation is an equation of vectors which is when solved, gives the result in the form of a vector.” In this topic, we shall […]

Implicit differentiation – Definition, Process, and Examples Implicit differentiation allows us to differentiate expressions (usually within an equation) that contain two or more variables. In our discussion, we will focus on implicitly differentiating equations with two variables. This technique is in fact an extension of the chain rule and you’ll learn why in our discussion. […]

Related rates – Definition, Applications, and Examples Learning how to solve related rates of change problems is an important skill to learn in differential calculus. This has extensive application in physics, engineering, and finance as well. In our discussion, we’ll also see how essential derivative rules and implicit differentiation are in word problems that involve […]

Hyperbolic functions – Graphs, Properties, and Examples The forms of hyperbolic functions (or hyperbolic trigonometric functions) may appear new but their properties are concepts and functions we’ve already encountered in the past. Hyperbolic functions were first studied by mathematicians like Johann Bernoulli to observe the behavior of the curve formed by a hanging chain. Any […]

Fundamental Theorem of Calculus – Parts, Application, and Examples From its name, the Fundamental Theorem of Calculus contains the most essential and most used rule in both differential and integral calculus. This theorem contains two parts – which we’ll cover extensively in this section. The new techniques we’ll be learning depend on the idea that […]

Sphere|Definition & Meaning Definition A sphere is the 3D counterpart of a circle and looks like a round ball. It has no edges, vertices, or face (just a curved surface). Like a circle in 2D, every point on the sphere’s surface is equidistant from its center, meaning it is perfectly symmetrical. In fact, if you […]

Conic Sections – Types, Properties, and Examples Conic sections are important mathematical concepts – even the ancient Greek mathematicians were fascinated with these three special curves. Conic sections have numerous applications in science and technology, including optics, astronomy, and even architecture. Conic sections are the result of intersecting the surfaces of a cone (normally, a […]

Polar to rectangular equation – Equations, Graphs, and Examples We can convert polar equations to rectangular form to rewrite a rectangular equation in terms of $x$ and $y$ to an equation of the form $r$ and $theta$. Knowing how to convert equations to rectangular and polar forms will help observe multiple relationships between two sets […]

Parametrize a line – Equations, Graphs, and Examples We can parametrize lines and line segments to understand the initial and ending positions of objects that we are observing. Learning about the steps of parametrizing a line can help describe the motion of an object or the behavior of the object given the third parameter. We […]

Vector multiplication – Types, Process, and Examples Vector multiplication helps us understand how two vectors behave when combined. This vector operation has an extensive application in physics, engineering, and astronomy, so we need to learn about these techniques, especially if we study higher maths. Vector multiplication covers two important techniques in vector operations: scalar product […]

Frequency Distribution – Definition and Examples A frequency distribution is any graph or table that shows the frequency of a set of data. Graphical displays of frequency include histograms, dot plots, and stem and leaf plots. There are also different tabular displays, including joint frequency tables and listed tables. Statistics employs frequency distributions for data […]

Pie chart – Explanation & Examples The definition of the pie chart is: “The pie chart is a circular chart used to represent categorical data using circle slices” In this topic, we will discuss the bar graph from the following aspects: What is a pie chart? How to make a pie chart? How to read […]

Evaluating Trig Functions – Explanation and Examples Evaluating trig functions exactly and without a calculator involves memorizing a few trigonometric values and identities. Knowing just the trig ratios in the first quadrant is enough to find angles in other quadrants with just a few tools. While calculators can give decimal answers, these methods give exact […]

Inverse Trigonometry-Explanation and Examples Inverse trigonometric functions use trigonometric ratios to find angles. There are two ways to write trigonometric functions. The first is to write $fun^-1x$, and the other is $arcfunx$, where $fun$ can be any of the six trigonometric functions. Just like standard trigonometric functions, inverse trigonometric functions are important in geometry and […]

Quadrantal Angles – Explanation and Examples Quadrantal angles are angles whose terminal ray lies along one of the two axes in a unit circle. Therefore, all quadrantal angles are whole number multiples of a right angle. Quadrantal angles play an important role in trigonometry, and applications extend to physical sciences and engineering. What Is a […]

Trigonometric Ratios – Explanation and Examples Trigonometric ratios are the ratios of certain sides in a right triangle given an angle measure. These trigonometric ratios are the trigonometric functions sine, cosine, tangent, and their inverses. Each function represents a different ratio in a right triangle given an angle measure. Trigonometric ratios are important in trigonometry, […]

Reference Angle – Explanation and Examples A reference angle is an acute angle between a given angle’s terminal ray and the x-axis. Reference angles are always positive, and an angle’s reference angle can be itself. Reference angles are useful in trigonometry and sciences that make use of it, such as astronomy, architecture, and engineering. Before […]

Coterminal Angles-Explanation and Examples Coterminal angles are angles that have the same terminal side. The terminal side is the second ray on an angle. These angles look the same on a plane, but their measure may not be the same because of the measuring direction or because the angle represents more than one circuit. Coterminal […]

Power Function – Properties, Graphs, & Applications Ever worked with a function that contains a single term? Chances are, you’ve been working with a power function. This function type is so diverse that if you’re studying functions, we’re 100% sure that you’ve already encountered a type of power function without knowing that it is one. […]