Advent calendar 2021
5 December
How many different isosceles triangles are there whose perimeter is 50 units, and whose area is an integer number of square-units?
(Two triangles that are rotations, reflections and translations of each other are counted as the same triangle. Triangles with an area of 0 should not be counted.)
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The largest possible isosceles triangles with a perimeter of 50 units is the equilateral triangle with sides of length 50/3. The area of this triangle is 120.28 square-units.
By continuously adjusting the width of the base of the isosceles triangle, triangles with every area between 0 and 120.28 can be created. The animation below shows this.
Triangles with areas of each integer from 1 to 120 can therefore be created. Each area can be made twice: once with a tall and thin triangle, and once with a short and wide triangle.
This means that there are 240 different triangles with a perimeter of 50 units and an integer area.
![](javascript:showlimage("advent2021-isosceles.gif"))