The total number of of triangles in a duo-decagon is
A duo-decagon (also known as a dodecagon) is a 12-sided polygon. The formula to find the number of triangles in a polygon is n(n-3)/2.
So, if n = 12, the equation becomes
12(12-3)/2 = 12(9)/2 = 54
There are 54 triangles in a duo-decagon.
To calculate the total number of triangles in a duo-decagon, we need to consider the possible combinations of three vertices from the twelve vertices of the shape.
A duo-decagon has 12 vertices, denoted by V₁, V₂, V₃, ..., V₁₂.
To form a triangle, we need to select 3 vertices from this set.
The number of ways to choose 3 vertices from a set of 12 can be found using the formula for combinations:
C(n, k) = n! / (k!(n-k)!)
Where n is the total number of items and k is the number of items to choose.
In our case, n = 12 (number of vertices) and k = 3 (number of vertices needed to form a triangle).
C(12, 3) = 12! / (3!(12-3)!)
= 12! / (3!(9)!)
= 12 × 11 × 10 / (3 × 2 × 1)
= 220
Therefore, there are 220 triangles in a duo-decagon.