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Browse High School Triangles and Other Polygons
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Area of an irregular shape.
    Classifying quadrilaterals.
    Heron's formula.
    Polygon diagonals.
    Pythagorean theorem proofs.
    Triangle congruence.



How Did Eratosthenes Measure the Circumference of the Earth? [6/26/1996]
Didn't Eratosthenes measure the lengths of shadows of sticks at different locations on the same day and time of the year, so he had two right triangles...?

How Long is the Hypotenuse? [07/12/1999]
In a right triangle, the lengths of the segments connecting the points of trisection of the hypotenuse to the vertex of the right angle are 7 and 9...

How Many Congruent Triangles? [11/11/2001]
Given a scalene triangle and a point P on some line L, how many triangles are there with one vertex at P, another vertex on L, and each triangle congruent to the given triangle?

How Many Proofs of the Pythagorean Theorem? [03/27/2003]
Do you know the exact number of proofs of the Pythagorean Theorem in existence?

How Many Triangles Have Sides Whose Lengths Total 15? [5/6/1995]
How many triangles can one construct with integral sides adding up to 15?

How Much Material to Purchase? [11/01/1997]
Sanchez warehouse wants to install a 3-foot wide ramp from the level floor to the top of the 4-foot high platform...

How Tall is Hal? [04/18/2001]
Hal is standing 40 feet away from a 36-ft. tree. If the distance from the top of the tree to the top of Hal's head is 50 ft., how tall is Hal?

How to Build a Proof [05/18/1999]
Given: Triangle ABC is a right triangle... Prove: Angle A and angle B are complementary angles.

How to Increase the Sides to Double the Area of a Square [01/28/2005]
I am puzzled as to how you would double the area of a square. Do you multiply the sides by some decimal?

How to Name a Polygon by Vertices [06/30/2004]
Is there a specific convention for naming the vertices of polygons? For example, picture a triangle with vertices R, S, and T. Would you call this triangle RST or RTS? In other words, is the figure named by going clockwise or counter-clockwise? How do you determine the first vertex named?

Importance of Reasonable Approximation [08/07/1999]
A stairway profile, and the calculation of arc length and curved surface area.

Importance of Surface Area [05/26/2001]
Why is surface area so important? What kinds of things depend on surface area?

Impossibility of Constructing a Regular Nine-Sided Polygon [04/07/1998]
Can you construct a regular 9 sided polygon with just a compass and straightedge?

Incenter and Conway's Circle [12/17/2002]
In a triangle, the bisectors of the angles intersect at a point in the interior of the circle. If I use this point as a center to draw a circle, what is the relation of this circle to the triangle?

The Incenter and Euler's Line [11/27/2001]
Why is the incenter of a triangle not on the Euler line?

Incenter Equidistant from Sides of Triangle [11/18/2001]
Prove that the point of intersection of the angle bisectors of a triangle is equidistant from the sides of the triangle.

Incenter, Orthocenter, Circumcenter, Centroid [01/05/1997]
I have been having trouble finding the Euler line of a triangle.

Incenters, Orthocenters, and the Spieker Point [02/13/2000]
Prove that the circumcenter of a triangle is the orthocenter of its medial triangle, and that the incenter of the triangle is the orthocenter of the triangle formed by the 3 excenters.

Incircles Tangent to a Common Line [03/23/2001]
In triangle ABC, the incircle touches side AB at M. T is an arbitrary point on BC. How can I show that the incircles of triangles BMT, AMT and ATC are all tangent to a common line?

Inclusive and Exclusive Definitions [04/05/2001]
Are squares rectangles? Are rectangles squares?

Inclusive Definitions: Trapezoids [11/04/2004]
As far as I know, a trapezoid is defined as a quadrilateral with exactly one set of parallel sides. However, a very highly regarded educator and textbook author recently argued that this definition is incorrect. His definition of a trapezoid is that it is a quadrilateral that has at least one pair of parallel sides. A square, therefore, would be considered a trapezoid. Is he correct or are thousands of books going to be published with the wrong definition?

Inclusive vs. Exclusive Definitions [01/24/2002]
My geometry teacher says that a square is not also a rhombus, a rectangle, and a parallelogram. Please help!

Inconstructible Regular Polygon [02/22/2002]
I've been trying to find a proof that a regular polygon with n sides is inconstructible if n is not a Fermat prime number.

Incribing a Pentagon in a Circle [2/6/1996]
I'm stuck trying to inscribe a pentagon. I can easily inscribe a square by just drawing two perpendicular diagonals. I also know that 360/5 = 72 but that doesn't help me at all. Can you help?

Inscribed, Circumscribed Circles [04/25/2003]
Given three general points in a plane of coordinates, (a,b), (c,d), and (e,f), what are the equations of the circles circumscribed about and inscribed within the triangle they form?

Inscribing a Regular Pentagon within a Circle [04/15/1999]
What are the reasons for the steps in inscribing a regular pentagon within a circle with only the help of a compass and a straightedge?

Inscribing a Square in a Triangle [10/13/2000]
How do you inscribe a square in a scalene triangle?

Inscribing a Square within a Half-Circle [04/05/2002]
Is there a way to inscribe a square within a given half-circle?

Insufficient Altitude [10/09/2003]
The base of a triangle is 1200 ft. The altitude is 500 ft. What is the length of the third side?

Integral of Triangular Surface [8/9/1996]
Is it possible to numerically integrate S { 1/3 (x^3 i + y^3 j + 0 k) . n} dS where n is the unit normal to the surface S, a triangle in a plane?

Interior and Exterior Angles [10/19/2001]
How can the sum of the angles in my quadrilateral be 280 degrees?

Interior Angles of a Polygon [10/21/1996]
The sum of the measures of the interior angles of any convex polygon with n sides is (n-2)180 degrees. Does this theorem apply to concave polygons?

Interior Angles of a Polygon [05/20/1997]
How do you figure out the sum of the interior angles of a polygon?

Intersection of Angle Bisectors of Triangles [02/17/1998]
Prove that bisectors of each angle of a triangle intersect at one point.

Intersections of Bisectors [9/6/1995]
Explain how to get the incenter, circumcenter, and orthocenter of a triangle.

Inverse Pythagorean Theorem [05/10/2001]
How can you tell a triangle is a right triangle without looking at the triangle and just how long the sides are?

Inverse Sine of a Value Greater than One [12/22/2003]
If I know that sin(B) = 1.732, why can't I find angle B? When I try to use my calculator it says, 'Error'. Does this have to do with the sine curve or is it something else?

Irregularly Inscribing a Circle ... of What Radius? [12/24/2011]
A student struggles to determine the radius of a circle given the sidelengths of an inscribed irregular polygon. First, Doctor Greenie offers a numerical approach; then, Doctor Floor follows up with the desired analytical method, applying Ptolemy's theorem and the rational root theorem to a key insight: the radius of the circumcircle remains invariant under a re-ordering of the polygon's sidelengths.

Is a Circle a Polygon? [03/07/1999]
A circle is not a polygon.

Is an Equilateral Triangle Also Isosceles? [02/19/2008]
Can equilateral triangles also be classified as isosceles? In other words, does an isosceles triangle have exactly two equal sides or at least two equal sides?

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