What is Radius: Definition and 1000 Discussions

In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin radius, meaning ray but also the spoke of a chariot wheel. The plural of radius can be either radii (from the Latin plural) or the conventional English plural radiuses. The typical abbreviation and mathematical variable name for radius is r. By extension, the diameter d is defined as twice the radius:






{\displaystyle d\doteq 2r\quad \Rightarrow \quad r={\frac {d}{2}}.}
If an object does not have a center, the term may refer to its circumradius, the radius of its circumscribed circle or circumscribed sphere. In either case, the radius may be more than half the diameter, which is usually defined as the maximum distance between any two points of the figure. The inradius of a geometric figure is usually the radius of the largest circle or sphere contained in it. The inner radius of a ring, tube or other hollow object is the radius of its cavity.
For regular polygons, the radius is the same as its circumradius. The inradius of a regular polygon is also called apothem. In graph theory, the radius of a graph is the minimum over all vertices u of the maximum distance from u to any other vertex of the graph.The radius of the circle with perimeter (circumference) C is





{\displaystyle r={\frac {C}{2\pi }}.}

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  1. Vanadium 50

    I Spherical trig - sphere radius from 6 lengths

    Four points lie on the surface of a sphere. Given the six distances between the points, calculate the radius of the sphere. This is (allegedly) an advanced high school level problem. However, it is a remembered problem, so it is possibly misremembered (i.e. there might have been some “bice...
  2. J

    Find at what rate the orbit radius will grow

    In order to change the radius, additional energy is required, the total energy of mass m on a circular orbit is given by: $$E_{total} = - \frac {GMm} {2R},$$ The change in energy between orbits ##R## and ##R_{0}## is: $$\Delta E_{total} = \frac {GMm} {2} \cdot \left( \frac 1 R_{0} - \frac 1 R...
  3. J

    Concerning the 3d surface of a Hypersphere

    Although 'undefined' it may be reasonable to assume that the hypersphere radius and circumference will respond to the normal relationship of a sphere i.e. 2 times Pi times r. Reasonable I suggest because the 3d surface of a hypersphere is a sphere. To get to my point, time and distance can be...
  4. wanwa

    Discover the Correct Star Radius with These Key Answers | Choose Wisely!

    I have the key answer for this (choosing 1 answer) A. 0,018 B. 1,134 C. 0,278 D. 0,974 E. 0,982
  5. E

    Pearson HW23, The Little Prince

    A. Correct answer is radius = 1770m, acceleration = 2.73*10^-3m/s. B. I don't know how to approach this problem. I don't know if I should start with forces, energy, or basic kinematics.
  6. T

    Speed of a mass falling into a star given the mass and radius of the star

    I tried the square root of ((2)(6.67*10^-11)(3.90E+30))/(5.70E+7) I got 1.55*10^-5 and that is wrong. Maybe I am using the wrong equation but this is the one of professor gave me and I don't know what I am doing wrong :-(
  7. L

    Electric field near conducting shell

    How would I do this question? I am having trouble figuring out what the radius is meant to be, why is it not 3R?
  8. FEAnalyst

    Beam deflection and curvature radius formula doubts

    Hi, I am working with leaf springs and studying the derivation of the formula for the deflection of such a structure. The derivation is shown here: My only doubt is how to obtain the following formula: $$\delta=\frac{L^{2}}{8R}$$ where: ##\delta## - deflection, ##L## - length of the beam...
  9. F

    Determining Radius from Magnetic Field of a Single-Wire Loop

    So I thought I knew how to do this problem but I've run into some issues that make the algebra feel impossible and I am beginning to feel like I'm taking the wrong approach, I ended up rewriting it in a doc because I was concerned maybe my handwriting was the cause of my error so the work is...
  10. neroE

    Electric Field due to a disk of radius R in the xy-plane

    Hello, This question, which I found in various electricitiy and magnetism books (e.g. Introduction to electrodynamics grif.). There are many variations of this question, I am mainly interested in the following setup of it: -Suppose there is a charged disk of radius R lying in the xy-plane, and...
  11. Ivan Seeking

    Schwarzschild radius of the Universe

    https://www.msn.com/en-us/news/technology/the-astonishing-scientific-theory-that-says-the-universe-might-be-inside-a-black-hole/ar-AA17lxtF?ocid=msedgdhp&pc=U531&cvid=272cb184de9c48fbbd3b321120e37dac Michio Kaku has often joked, "If you want to know what it looks like inside of a black hole...
  12. C

    Calculating Radius & Water Depth of a Spherical Bowl

    The water level in a spherical bowl has a diameter of 30 cm. If the horizontal diameter of the bowl is 10 cm below the water level, calculate the radius of the bowl and the depth of the water in the bowl. I managed to draw a diagram below: In my drawing, I am seeing the sphere ABCD as the...
  13. S

    B Calc Excess Rad on Earth in Curved 3D Space | Feynman Lectures 6-2

    I'm reading the Feynman lectures chapter on "Curved Space", section 6-2. Say we're trying to figure out a way to measure average curvature on Earth. We know that 3d space is curved if Euclidean geometry rules don't work - e.g. the ratio of circumference and radius of a circle isn't ##2\pi##, or...
  14. uSee2

    Circular Motion with Decreasing Radius

    The answer key states that the new tangential speed is half the original speed. However, this isn't correct right? It should double. My proof: ##F_c = \frac {mv^2} R## ##F_c = F_t## ##\frac {mv^2} {\frac R 4} = \frac {m(2v)^2} R## If centripetal force were to stay constant. As such, tangential...
  15. F

    A Orbital Radius of a Jovian Moon

    How did the first scientists figure out the orbital radius of a moon of Jupiter? How can observations lead to a calculation of the orbital radius of a moon of another planet?
  16. A

    Angular Velocity from KE, radius, and mass

    I tried using the equation w^2 = (4*K)/(mr^2) but I don't think this is right... I got my answer to be 3.2243 and that's not correct
  17. C

    I Radius of Gyration: Definition & Uses

    What is the radius of gyration and how is it used?
  18. chwala

    Find the rate of increase of the radius of the pool

    My approach is as follows; ##\dfrac{dr}{dt}=\dfrac{dr}{da}⋅\dfrac{da}{dt}## ##\dfrac{dr}{da}=\dfrac{1}{2πr}## ##\dfrac{dr}{dt}=\dfrac{1}{2πr}⋅15## ##\dfrac{dr}{dt}=\dfrac{7.5}{πr}## and noting that ##r= \sqrt{\dfrac {50}{π}}## ##\dfrac{dr}{dt}=\dfrac{7.5}{12.533}=0.598## cm/s *kindly note...
  19. V

    Significant digits rule when determining radius from diameter

    Probably, to satisfy the significant digits rule for division, we should consider ##r = 5.0 \div 2.0##. But I'm unable to come up with a reason why significant digits rule should not apply to ##r= d \div 2##. Also, if we apply significant digits rule to this calculation then we loose accuracy...
  20. A

    I Differences b/w Schwarzschild Radius & Event Horizon

    I understood that the event horizon is a null surface and not a place in space, what is the relationship between it and the Schwarzschild radius? Also, what does the Schwarzschild radius physically represent for example for an object such as a star?
  21. D

    Estimating the Bohr radius from the uncertainty principle

    Soo. I think this problem is too direct and easy so I think I got it in wrong way: p=h/r and then plug in the K and V and then we get E=E(r) and get derivative and we have minimum? What do you think? is there sth I am missing?
  22. A

    MHB Formula to find arc radius using arc length, chord length, and/or segment angle

    I'm attempting to write some code using the Ruby programming language that will give me the radius of an arc but the only pieces of information I have to work with are the arc length (L), the chord length (C), and the circular segment angle in degrees (A): I'm hoping someone can show me how to...
  23. A

    Find the inertia of a sphere radius R with rotating axis through the center

    $$I = \int{r^2dm}$$ $$dm = \sigma dV$$ $$dV = 4\pi r^2dr$$ $$\sigma = \frac{M}{\frac{4}{3}\pi*R^3}$$ $$I = \sigma 4 \pi \int_0^R{r^4 dr} = \frac{3*MR^2}{5},$$ which is not the correct moment of inertia of a sphere
  24. A

    Find the center of mass of a uniform semicircular plate of radius R

    $$rcm = \frac{1}{M}\int_0^\pi(rdm)$$ $$dm = \sigma{dA}$$ $$dA = (\pi*R^2)*\frac{d\theta}{2\pi}$$ $$\sigma = \frac{M}{\frac{\pi*R^2}{2}}$$ $$dm = M*\frac{d\theta}{\pi}$$ $$r = R(cos(\theta)\vec i + sin(\theta)\vec j)$$ $$rcm = \int_0^\pi{\frac{R}{\pi}(cos(\theta)\vec i + sin(\theta)\vec j)} =...
  25. JD_PM

    A Cavitation: radius of a bubble for compressible flow

    The liquid-vapor mass transfer (evaporation and condensation) is governed by the vapor transport equation: $$\frac{\partial}{\partial t} (\alpha_l \rho) + \nabla \cdot (\alpha_l \rho \vec v) = \dot m^{+} + \dot m^{-}$$ In the incompressible flow case (constant density), it reduces to $$...
  26. chwala

    Find the radius of the smaller circle in the tangent problem

    Find the question here and the solution i.e number 10 indicated as ##6-4\sqrt{2}##, I am getting a different solution, my approach is as follows. I made use of pythagoras theorem for the three right angle triangles as follows, Let radius of the smaller circle be equal to ##c## and distance...
  27. G

    B Do I have enough information to find the radius?

    If there is a circle, 360 degrees, and 2 random points A & B are chosen on the circle. A straight line is drawn between those two points. You are not shown the whole circle. All you can see is A, B the straight line and an arc representing an unknown number of degrees. You know the length of the...
  28. L

    Calculating the Radius of a Hot Air Balloon to Withstand a Load of 300 kg

    The result is supposed to be 12,2 m but every time I get 8,016 m... I used for example this formula >r=m/[(density of air-density of hot air)*(4/3)*pi] For density I used > rho=(p*M)/(R*T) Am I forgetting something? Thanks in advance.
  29. S

    Why are v and a decreasing as the oribit radius increases?

    I don't understand part (c) the answer is decreasing for both. I would have it decreasing for the centripetal acceleration and then increasing for the velocity, why is that not correct? Thanks in advance!
  30. Buzz Bloom

    I Please check my calculation of the radius of a finite universe

    My source for Ωk and H0 is https://www.cosmos.esa.int/documents/387566/387653/Planck_2018_results_L06.pdf , page 40, equation 47b. Ωk = 0.0007. Page 15 equation 13 has H0 = 67.27 km/(s Mpc). The formula I used is R = (c/H0) (1/Ωk)1/2 . ( I did have a reference for this, but I misplaced it, and I...
  31. A

    The radius of convergence of a series

    Greetings! I have a problem with the solution of that exercice I don´t agree with it because if i choose to factorise with 6^n instead of 2^n will get 5/6 instead thank you!
  32. Send-Help

    Find tangential velocity given radius and the coefficient of friction

    I have attempted to solve for the velocity by setting the centripetal force (mv2)/r to the normal force pointed to the center of rotation (mg). This approach seems to give the incorrect solution and I am unsure of my misunderstandings.
  33. S

    Bohr radius of Earth-Sun system

    When I looked up for Bohr radius, the formula has ##q## in it, which is charge of the object. For this question, the electron and proton are replaced by sun and Earth so it means that I have to know the charge of Earth and sun? Thanks
  34. bluesteels

    Quick question about which radius to use on Gauss' law problem

    confused on part A/B when I look up they did E= Q/2e(0.8)^2. But why not use the 0.100mm because that is the area of the enclosed. Same with B why did they use 100m and not 0.8m because 0.8 is smaller so it enclosed the charge
  35. tanaygupta2000

    Limiting radius ratio for tetrahedral

    I am able to prove that it is 0.225 but how do I prove that it is also 0.414? I need to find the max. and min. packing fraction values, which I got as a function of (r1/r2) Please help
  36. F

    Engineering Calculating the flux through a certain radius (concentric charged spheres)

    Question My attempt for the 1st part, when r = 1.5 m Could someone confirm my answer?
  37. F

    Engineering Calculating the flux through the spherical surfaces at certain radius

    Relevant Equation: My attempt: Could someone please confirm my answer?
  38. S

    Straight wire inductance vs wire radius

    I know that the whole topic of inductance in a straight wire is complicated (and has led to some heated discussions in this forum :smile:). I followed Rosa's derivation and can see that it leads to an inverse relation of the inductance to the wire radius, and from what could understand, the...
  39. desperatestudent123

    Calculating magnetic field given dl, current, and radius vector

    I used the above equation, and started with getting the cross product of dl and r, which was equal to 0.00195i+0.00365k. From there, I divided each component by the magnitude of radius cubed (0.827^3). I then multiplied by I and u naught(u_0=4pi*10^-7), and then divided by 4pi. The answer I got...
  40. E

    How Do You Calculate the Radius of a Wire Using Electrical Resistance Formulas?

    i really have tried all the formulas out there and can't seem to get a solid answer
  41. A

    Finding the Radius of Convergence for Y=6x+16 - Troubleshooting and Solution

    Greetings I have some problems finding the correct result My solution: I puted Y=6x+16 so now will try to find the raduis of convergence of Y so let's calculate the raduis criteria of convergence: We know that Y=6x+16 Conseqyently -21/6<=x<=-11/6 so the raduis must be 5/3. But this is not...
  42. T

    A Radius of the Sun: Can We Accurately Predict It Through Theoretical Models?

    Basically, I'm wondering if there have been any attempts to calculate/model what the radius of the sun should be based on gravitational, thermal, and electromagnetic pressures. If there has, where can I find the calculation/model, and how closely does it match the actual radius of the sun...
  43. WMDhamnekar

    MHB How is Radius of Curvature Computed for a Given Curve?

    How did the author compute the highlighted term 2 from the highlighted term 1 in the following answer to the given question? If $\rho =\frac{d\psi}{ds}$, then the term 2 should be $\upsilon^2 \frac{d\hat{T}}{d\psi}\rho$, but instead, it was written...
  44. karush

    MHB -2.4.27 find center and radius of circle

    Determine the graph of $x^2+y^2+6x+8y+9=0$ $\begin{array}{rll} \textsf{rewrite} &(x^2+6x )+(y^2+8y)=-9\\ \textsf{complete square} &(x^2+6x+9)+(y^2+8y+16)=-9+9+16\\ \textsf{simplify equation} &(x+3)^2+(y+4)^2=16=4^2\\ \textsf{observation} &C(-3,-4), \quad R=4...
  45. T

    Heat Transfer - Critical Radius of Insulation

    A steel pipe with foam insulation is embedded in a concrete wall. The steel pipe is carrying cold water and therefore gains heat from outside atmosphere. Heat transfers through the concrete wall, foam insulation and then to the pipe. Is it possible to calculate critical radius of insulation for...
  46. momoneedsphysicshelp

    How do mass and radius affect gravitational pull on planets?

    Can someone please verify if my reasoning is accurate? I chose E) Planets B and D because they both have the same ratio of mass to radius which is the lowest of all the other planet options. Due to the fact that they have mass and radius evened out the gravitational pull will pull weight down...
  47. H

    The net current through a circle of radius R, in the xy plane and centered at the origin is given by?

    Here's what I did: ∮ B * dl =μ0 * I ∮ AR * 2π*R =μ0 * I ∮ 2π*AR^2 / μ0 = I ∮ 2π*AR^3 / 3μ0 = I Where did I do wrong?
  48. Lilian Sa

    Star collapse in general relativity — pressure as a function of star radius

    What I've done is using the TOV equations and I what I found at the end is: ##e^{[\frac{-8}{3}\pi G\rho]r^2+[\frac{16}{9}(G\pi\rho)^{2}]r^4}-\rho=P(r)## so I am sure that this is not right, if someone can help me knowing it I really apricate it :)
  49. H

    MHB Calculate the radius of the circle

    A, B and C are points on a circle with center O. Angle ABC = $75°$ . The area of the shaded segment is $200cm^2$ . Calculate the radius of the circle. Answer correct to $3$ significant figures.
  50. A

    Problem in finding the radius of convergence of a series

    Good day I'm trying to find the radius of this serie, and here is the solution I just have problem understanding why 2^(n/2) is little o of 3^(n/3) ? many thanks in advance Best regards!