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Browse High School Conic Sections, Circles
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    About ellipses.
    Find the center of a circle.
    Is a circle a polygon?
    Volume of a tank.
    Why is a circle 360 degrees?

The Perimeter of an Ellipse or Oval [9/5/1996]
What is the formula for calculating the perimeter of an oval, or an ellipse?

Perimeter of an Inscribed Regular Polygon [12/10/1998]
What is the formula for the perimeter of a regular polygon inscribed inside a circle?

Perimeter of an Oval [04/08/1999]
What is the perimeter of an oval called?

Perimeter of a Right Triangle [08/14/2001]
What is the perimeter of a right triangle with hypotenuse 65 that can be circumscribed about a circle with radius 12?

Placing Coins That Touch [8/7/1996]
How many 20-cent coins can you put around a 20-cent coin so that all of them touch?

Point in a Circle [04/29/1997]
Given a circle with two 6-inch chords running across the top and the bottom... find the probability that a point chosen at random is in the region between the chords.

Point on an Ellipse [05/16/1997]
Given an ellipse and an arbitrary angle theta from either axis, how do you find the coordinates of the intersection of the ellipse and a vector formed by angle theta?

Points within an Ellipse [06/03/2003]
How to generate points that will be within an ellipse.

Polygons, Infinite Sides, and Circles [04/03/1997]
Can a regular polygon with an infinite number of sides be a circle?

Probability That Random Chord Exceeds Radius in a Circle [11/20/2004]
If a chord is selected at random on a fixed circle, what is the probability that its length exceeds the radius of the circle?

Product of Radii of Two Circles [7/22/1996]
The length of a common internal tangent to two circles is 7, and a common external tangent is 11...

Product of the radii [7/3/1996]
The length of a common internal tangent to two circles is 7, and a common external tangent is 11...

Proof of the Feuerbach Theorem [03/14/2000]
Please submit the proof of the Feuerbach theorem (the nine-point circle is tangent to the incircle and the circumcircle of a triangle.)

A Proof using Analytic Geometry [02/24/1999]
Prove that, if p is a point inside the ellipse, there is one and only one chord QP bisected at P.

Putting a Ribbon around the Earth [12/23/2006]
I've heard that if you have a ribbon wrapped around the equator of the earth, and you want to increase its length so that it floats 1" above the earth all the way around, you only need to add 6.28" to the ribbon. Is that really true? How is it possible?

Quadrilaterals and Inscribed Circle [05/06/1999]
From ten sticks of lengths 1,2,3,....,10 four are selected to form the sides of a quadrilateral...

Radius and Center of a Circle from 3 Points [07/23/1999]
Given the coordinates of three points on a circle, how can you find the center and radius?

Radius, Center of Circle Given 3 Points [8/28/1996]
What is an easy formula to calculate the center point and radius of a circle given three points on the circumference?

Radius from an Arc and a Chord [06/08/1999]
If I know the height of an arc from the midpoint on a chord, and the length of the chord, can I find the radius of the circle of which the arc is a part?

Radius of a Circle Inscribed in a Triangle [06/02/1999]
What is the radius of an inscribed circle of a triangle with sides 3, 4, and 5?

Radius of an Arch [05/25/1998]
Is there a formula to calculate the radius of a circle given the chord length and the distance from the centre of the chord to the circle?

Radius of a Racing Circle [09/20/1999]
How can I find the equation for the radius of a 'racing circle' (the fastest path a racecar can take through the corner defined by the quadrants of two circles), an arc sandwiched between identical quadrants of two concentric circles?

Radius of Circumscribed Circle [05/11/2001]
Where can I find a derivation of R = abc/4K?

Radius of Curvature [07/23/2003]
I have several ellipses whose major and minor diameters I know, but I have no information about their foci.

Rail Bend in Hot Weather [10/13/2002]
A 20-ft piece of rail expands 1 in. in length during a hot spell. If there are no expansion gaps, how high off the ground will the rail rise?

Railroad Track Expansion [04/07/2003]
A continuous straight railroad track of one mile is permanently tied down at both ends. As the day heats up, the coefficient of expansion of steel causes the rail to expand so that the length is now 5281 feet. Assuming that the track expands upward, what maximum vertical distance from the horizontal will the track rise at the highest point?

Ratios, Geometry, Trigonometry [06/10/1999]
A homeschool teacher asks for help with triangles, flagpoles, and circles.

Rectangular Hyperbola [06/11/2002]
The graph of a rectangular hyperbola looks nothing like a rectangle. Where does the name come from?

Reflection Points on a Circle-Shaped Mirror [09/30/2003]
Points A and B are located within a circle. If A were a light emitting point and B a light receiving point, then B would receive light from points P on the circle. How can I find these points?

Reflections in Parabolic Mirrors [05/14/2005]
I know that in a parbola, any ray that starts at the focus and hits the parabola is reflected parallel to the central axis of the parabola. Can you explain or prove why that happens?

Reflective Properties of a Semicircular Mirror [05/16/2000]
What are the reflective properties of a semicircular mirror? Will a ray exit a semicircular mirror parallel to its entry line?

Relationship between Circumference and Area [11/08/2007]
My 6th grade daughter asked if there is a shortcut for converting from area to circumference and back without finding the radius.

Rhumb Lines and Great Circle Routes [09/24/1998]
Can you explain great circles and rhumb lines and how they relate to shortest distances in geometry?

Right Triangle Inscribed in a Parabola [09/20/1999]
Show that the point of intersection Q of the axis of the parabola y=x^ 2 and the hypotenuse of right triangle RST (inscribed in the parabola so that R coincides with the vertex of the parabola) is independent of the choice of right triangle.

Rounding Pi [06/01/1999]
Can you prove that the value of Pi cannot be rounded down to 3.0? Is there an error in the Bible?

Scaling Separated Circles So That They Intersect [09/29/2014]
What common factor would scale up the radii of three non-intersecting circles so that they do intersect? Starting with generic circles, Doctor George outlines one approach and sets up the algebra.

Secant-Tangent Theorem [03/21/2002]
I'm trying to prove the secant-tangent theorem.

Semicircle Proof [02/27/2001]
M is the midpoint of AB. Three semicircles with diameters AM, MB, and AB are drawn. A circle with centre O and radius r touches all three. Prove that r = 1/6 AB.

Shared Points on Concentric Circles [03/11/2004]
Can two concentric circles share only a few points? If they are concentric and they have the same radius, they would share all of their points, and if they don't have the same radius they will share no points. It seems like it's all or none.

Simplifying Circle Formulas from the Dr. Math FAQ [04/30/2004]
In your FAQ on circle formulas, in the sections where the other five values are derived from any two known values, could you write each formula in terms of only the two known values, instead of using the intermediate steps?

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