Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [triangles]

For questions about properties and applications of triangles.

A triangle is a polygon with three sides. It is an important geometric figure, because any polygon can be subdivided into triangles.

Triangles can be classified by the number of sides they have that have equal length

  • All three sides of an equilateral triangle have equal length.
  • An isosceles triangle has at least two sides of equal length.
  • A scalene triangle is a triangle that is not isosceles, that is, it has no sides with equal length.

A triangle may also be classified by describing its angles. A triangle is said to be a right triangle if it contains a right angle, and obtuse triangle if it contains an obtuse angle, or an acute triangle if all three of its angles are acute.

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V.I. Arnold says Russian students can't solve this problem, but American students can -- why?

In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is 10 inches, the altitude dropped onto it is 6 inches. Find the area of the triangle. American…
Eli Rose
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Is the blue area greater than the red area?

Problem: A vertex of one square is pegged to the centre of an identical square, and the overlapping area is blue. One of the squares is then rotated about the vertex and the resulting overlap is red. Which area is greater? Let the area of each…
Mr Pie
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What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?

My friend gave me this puzzle: What is the probability that a point chosen at random from the interior of an equilateral triangle is closer to the center than any of its edges? I tried to draw the picture and I drew a smaller (concentric)…
terrace
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What is the most elegant proof of the Pythagorean theorem?

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite is this graphical one: According to…
Shane
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What's a proof that the angles of a triangle add up to 180°?

Back in grade school, I had a solution involving "folding the triangle" into a rectangle half the area, and seeing that all the angles met at a point: However, now that I'm in university, I'm not convinced that this proof is the best one (although…
Joe Z.
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Finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane? One approach is to find the length of each side from the coordinates given and then apply Heron's formula. Is this the…
TSP1993
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How to calculate the area of a 3D triangle?

I have coordinates of 3d triangle and I need to calculate its area. I know how to do it in 2D, but don't know how to calculate area in 3d. I have developed data as follows. (119.91227722167969, 122.7717056274414, 39.3568115234375),…
iamgopal
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Do two right triangles with the same length hypotenuse have the same area?

I watched computer monitors and I asked myself, do two monitors with the same display diagonal have the same display area? I managed to find out that the answer is yes, if two right triangles with the same length hypotenuse have the same area. The…
totymedli
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What is the name of this theorem of Jakob Steiner's, and why is it true?

In The Secrets of Triangles a remarkable theorem is attributed to Jakob Steiner. Each side of a triangle is cut into two segments by an altitude. Build squares on each of those segments, and the alternating squares sum to each other. The book…
MBP
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Is being a right triangle both necessary and sufficient for the Pythagorean Theorem to hold?

I recently encountered a Stack Overflow question (since closed) in which the OP was testing for whether a triangle was right by whether or not it "met" the criteria of the Pythagorean Theorem (i.e. whether or not the square of the hypotenuse is…
EJoshuaS - Stand with Ukraine
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Probability that 3 points in a plane form a triangle

This question was asked in a test and I got it right. The answer key gives $\frac12$. Problem: If 3 distinct points are chosen on a plane, find the probability that they form a triangle. Attempt 1: The 3rd point will either be collinear or…
Serenity
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Triangle inequality for subtraction?

Why is $|a - b| \geq|a| - |b|$?
CodeKingPlusPlus
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How many triangles are there?

The question is how many triangles are there in the following picture? I have thought to solve it by creating a formula based on the angles of the lines starting from the bottom of each side. I don't get it right though. Any clues/ideas would be…
Dimme
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Is every parallelogram a rectangle ??

Let's say we have a parallelogram $\text{ABCD}$. $\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two parallel lines $\text{AB}$ and $\text{CD}$, So, $$ar\triangle \text{ADC}=ar\triangle \text{BCD}$$ Now the things…
Harsh Kumar
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uniform random point in triangle in 3D

Suppose you have an arbitrary triangle with vertices $A$, $B$, and $C$. This paper (section 4.2) says that you can generate a random point, $P$, uniformly from within triangle $ABC$ by the following convex combination of the vertices: $P = (1 -…
dsg
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