find the number of triangles in a 15 sided figure if drawn from a single vertex
Answers
Answer:
Explanation:
Number the vertices with consecutive integers 1 through 15. Let the edge (side) of the polygon connecting vertex i and vertex i+1 be called edge
i
, for all i (
1
≤
i
≤
14
), and let the edge connecting vertices
15
and
1
be called
15
. Then, if you imagine walking clockwise around the periphery of the polygon, from vertex 1 back to vertex 1, you will turn to the right
15
times. The angles through which you have turned are the exterior angles of the polygon.
Since the polygon is regular, all sides and angles are equal, so each turn at each vertex is the same, and of size
360
°
15
=
24
°
degrees. That is, since you've returned to the starting point you must have completely turned around, i.e. turned
360
°
. Now, if you turn 15° to the right at each vertex, the amount you didn't turn, i.e. the size of the interior angle, is
180
−
24
=
156
°
.
Step-by-step explanation:
In any convex polygon, if you start at one vertex and draw the diagonals to all the other vertices, you will form triangles,
The number of triangles so formed is always
2
LESS than the number of sides. As each triangle has
180
°
, you can find the sum of the interior angles of the polygon:
For an
n
-sided polygon there are
(
n
−
2
)
triangles.
The sum of the interior angles is therefore
180
°
(
n
−
2
)
In a
15
-sided polygon:
Sum interior angles =
180
(
15
−
2
)
=
180
×
13
=
2340
°
Each interior angle of the regular polygon =
2340
°
15
=
156
°
Answer:
From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, and that the regular hexagon can be partitioned into six equilateral triangles.
Step-by-step explanation:
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