Answer:
80.55
Explanation:
Formula to calculate the perimeter (P) of a regular dodecagon with a radius (r) is:
P = 24 * r * sin(π/12),
where π is approximately 3.14159.
Using this formula, with the given radius of 13, we can calculate the perimeter as follows:
P = 24 * 13 * sin(π/12).
Calculating sin(π/12), we find that it is approximately 0.2588190451.
Substituting this value into the formula, we have:
P = 24 * 13 * 0.2588190451.
Evaluating the expression, we get:
P ≈ 80.55223123.
Therefore, the approximate perimeter of a regular dodecagon with a radius of 13 is approximately 80.55 units.
Perimeter = 12 × 13 × sin(π/12) ≈ 121.41 units
regular dodecagon with a radius of 13.
A regular dodecagon can be divided into 12 congruent isosceles triangles, each with a central angle of 30 degrees (360 degrees divided by 12 sides). The base of each triangle is a side of the dodecagon, and the two equal sides are radii of the dodecagon.
To find the length of the base of each triangle (i.e., the side of the dodecagon), we can use trigonometry. The formula is:
base = 2 * radius * sin(π/12),
where radius is 13 and π is approximately 3.14159.
Substituting the given values into the formula, we have:
base = 2 * 13 * sin(π/12).
Calculating sin(π/12), we find that it is approximately 0.2588190451.
Now we can substitute this value into the formula:
base = 2 * 13 * 0.2588190451.
Evaluating the expression, we get:
base ≈ 6.712685936.
Since there are 12 sides in the dodecagon, the perimeter is the sum of all the sides:
perimeter = 12 * base.
Substituting the value of the base, we have:
perimeter = 12 * 6.712685936.
Calculating the expression, we find:
perimeter ≈ 80.55223123.
Therefore, the approximate perimeter of a regular dodecagon with a radius of 13 is approximately 80.55 units
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