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Dr. Math FAQ:
Browse High School Euclidean/Plane Geometry
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Selected answers to common questions:
Pythagorean theorem proofs.
- Books about Proofs [10/26/1995]
A student asks for help with geometry proofs, and Dr. Math suggests two
- Bouncing Balls [11/14/1994]
Geometry Project - Problem: Balls bounce off of solid objects. Is there a
pattern to the bounce? Can you predict the bounce?
- Bretschneider's Theorem and Cyclic Quadrilaterals [11/30/2000]
Can you prove Bretschneider's Theorem for the area of a quadrilateral?
Also, can you show that any quadrilateral with supplementary opposing
angles can be inscribed in a circle?
- Building Two Column Proofs [09/12/1998]
We just started learning proofs, and I don't understand how to figure out
the ordering. Can you explain?
- Calculating a Mirror's Reflection [08/08/1999]
How can I find the angle and the point on a mirror to shine a light at in
order to illuminate an object?
- Can Two Curves Be Parallel? [12/19/2007]
Straight lines are parallel if they are equally distant and never
intersect. Can the graphs of quadratic or cubic equations be
considered parallel if they are equally distant and never intersect?
- Carpet and Room Areas [10/26/2001]
A man buys a roll of carpet 9 ft. wide by 12 ft. long to fit a 10ft. by
10 ft. room. When the roll of carpet is unrolled, a hole is discovered in
the middle of the carpet...
- Carpet Problem [10/15/2001]
You have to carpet a 9x12 room, but when you go the store they only have
a 10x10 carpet and a 1x8 piece of carpet...
- Catenary Curve [03/30/1999]
Find the vertex of a catenary curve.
- Centroid - Center of Gravity [03/25/2002]
Can a triangle have a unique centre of gravity?
- Choosing the Next Step in a Proof [11/03/2003]
How do I know which theorem is the correct one to use next when I'm
trying to prove something?
- Circle and Rectangle Area Problem [11/03/2003]
One corner of a rectangle is on the center of a circle. The radius is
larger than the small side of the rectangle but smaller than the large
side. What is the area of their intersection?
- Circle of Radius 1 km around Given Latitude/Longitude [08/27/2002]
I need feed points for a circle of radius 1 km with center at a given
latitude and longitude. Eight or more points should be enough for the
- Circumference vs. Perimeter [05/09/2003]
Technically speaking, can the term 'perimeter' apply to a circle in a
- Classifying Shape Based on Coordinate Points [01/03/2000]
How can I design an algorithm to classify shapes based on a relatively
small set of (x,y) coordinates that describe the boundary of a closed
- Cleaning the Ice [09/09/1997]
The hockey rink is a rectangle, 120 ft. by 60 ft. The scraper cleans a 4-
ft.-wide strip... on which trip will it have cleaned half the area of the
- The Collapse of Compasses that Do Not Copy Segments, and the Lengths We Go To [04/16/2011]
A teacher wonders, "Since when do collapsible compasses copy lengths?" Suggesting
that Euclid may have posited this basic ability early among his propositions -- for the
purposes of simplifying more interesting constructions -- Doctor Peterson then goes
on to discuss the pedagogical pros and cons of compass quality, reliance, and over-reliance.
- Collapsible Compass [11/21/2003]
What is a collapsible compass, and when would you use one?
- Comparing Sidelengths of Nested Triangles that Share the Same Base [09/07/2011]
A student struggles with a proof that starts with a triangle and a point in its interior.
Doctor Floor offers a boost by drawing a diagram, then invoking the triangle inequality
- Complementary and Supplementary Angles [09/30/1998]
Why are angles called complementary and supplementary?
- A Complete Proof about Tangential Circles [06/05/1998]
Can you show me a proof, with full justification, of the following
theorem? Two circles of the same radius touch at A ...
- Connected Sets in Topology [04/22/1998]
Exploring connected sets with examples in Euclidean space.
- Connection Between Circumference and Surface Area [05/08/1998]
Can you explain the connection between the circumference of a circle and
the surface area of a sphere?
- Constructing a One-Degree Angle [05/25/2000]
Is it possible to contruct a one degree angle using only a straightedge
- Converse, Inverse, Contrapositive [09/08/2001]
Write the converse, inverse, and contrapositive of each conditional and
determine whether they are true or false; if false, give a
- Converse of the Parallel Lines Theorem [05/28/2000]
How can I prove the converse of the Parallel Lines Theorem: If a
transversal intersects two lines so that the alternate angles are equal,
then the lines are parallel?
- Converses in Construction [12/31/2016]
A teen struggles with generating the converses of theorems. With examples aplenty,
Doctor Rick emphasizes context and validity.
- Converting QBasic Angles to Mathematical Angles [08/11/1999]
QBasic measures angles clockwise from north, while mathematicians measure
them counterclockwise from east. How can I convert QBasic's angle
measures to mathematical ones? Also, do negative angles exist?
- Counting Rectangles Cut By a Diagonal [06/15/1999]
How can we find an equation for the number of unit squares that are cut
by a line going from corner to corner on a rectangle?
- Counting Regions Formed by Straight Lines [04/18/1998]
How many regions are formed by n straight lines if no three meet in a
single point and no two are parallel?
- Covering Paper using Index Cards [10/24/2001]
What is the maximum area of an 8"x13" sheet of paper that you can cover
by using seven 3"x5" standard index cards?
- Cow Grazing in Circles [11/3/1994]
A cow is tied to a 100 ft. rope attached to a pole in the center of a
circle of radius 50 ft. This circle has a ten foot opening, out of which
the cow can walk to graze. What's the grazing area ?
- Defining a Square Centimeter [04/09/2002]
Why is the area of a unit square the product of the two sides?
- Defining Distance Mathematically [10/16/1996]
What is wrong with D' = sqrt(X^2 - X'2)?
- Defining Exterior Angles of Polygons [06/24/2004]
My math teacher says the sum of the exterior angles of a triangle is
900 degrees (360*3 - 180). I think that the sum is 360 degrees. Who
- Definition of Opposite Sides [01/18/2001]
What is the formal definition of 'opposite sides' of a polygon? Does a
regular pentagon have opposite sides? Does a concave polygon have
opposite sides? How can we define it consistent with our intuition?
- Definition of Oval [06/09/2002]
Can you give a precise definition of 'oval'?
- Definitions of Advanced Concepts [11/13/1998]
Can you give me definitions for: Pythagorean Triplets, Principle of
Duality, Euclid's Elements, Cycloid, Fermat's Last Theorem?
- Definitions of Edge and Face in 2D and 3D [10/10/2008]
What is the 'official' definition of 'edge'? Specifically, is an edge
restricted to the intersection of two non-coplanar faces or do two-
dimensional shapes have edges? I'm also curious about a definition of
'face'. How many faces does a two-dimensional shape have?
- Degrees and Radians, Explained [03/19/2002]
How do you find the degree measure for an angle from pi/60 rad?