# Tag Archives: equilateral triangle

## Equilateral Triangle and Regular Hexagon

What is the ratio of the area of an equilateral triangle to that of a regular hexagon inscribed in the same circle? Source: NCTM Mathematics Teacher, February 2008 Solution The hexagon is made up of six congruent isosceles triangles three … Continue reading

## Largest Equilateral Triangle in a Square

How long is the side of the largest equilateral triangle that can be inscribed in a square with side length ? Source: NCTM Mathematics Teacher, August 2006 Solution A first guess is an equilateral triangle of side length . Could … Continue reading

## Equilateral Triangle and Square

A pentagon made up of equilateral triangle with side length on top of square is inside a circle passing through points , and . Find the radius of the circle. Source: NCTM Mathematics Teacher, August 2006 Solution Let be the … Continue reading

## Two Equilateral Triangles

If two equilateral triangles of area intersect to form a regular hexagon, then what is the area of the hexagon? Source: NCTM Mathematics Teacher, September 2006 Solution If we subdivide an equilateral triangle into smaller congruent equilateral triangles as shown … Continue reading

## Arc of a Circle

is a diameter of a circle of radius unit. is a chord perpendicular to that cuts at . If the arc is of the circumference of the circle, what is the length of the segment ? Source: NCTM Mathematics Teacher, … Continue reading

## Equiangular Hexagon

An equiangular hexagon with consecutive side lengths , and can be inscribed in an equilateral triangle with side length . This same equiangular hexagon can also be inscribed in a second (distinct, noncongruent) equilateral triangle with side length . What … Continue reading

## Sum of Consecutive Numbers (Part 2)

The sum of consecutive positive integers is . Find all possible values of . Source: NCTM Mathematics Teacher, December 2005 SOLUTION Let be the sum of consecutive positive integers with the first term equal . If we add the first … Continue reading

## Measure of Angle

, and are vertices of a cube, as shown. Find the measure of . Source: NCTM Mathematics Teacher, December 2005 SOLUTION Triangle is equilateral because its three sides are diagonals of a square. The measure of equals . Answer:

## Tetrahedron

Four of the nonadjacent vertices of a cube form a regular tetrahedron. What is the ratio of the surface area of the cube to the surface area of the tetrahedron? Source: NCTM Mathematics Teacher, December 2005 SOLUTION Let the side … Continue reading

## Largest Value

The equilateral triangle has side length , and is a point on the line segment . Denote by the area of triangle and by the area of triangle . What is the largest value of the product ? Source: NCTM … Continue reading