Find the perimeter of a pentagon circumscribed about the circle that has a radius of 1.5 units.
consider the pentagon as 5 isosceles triangles, each with vertex angle of 72°, and sides of 1.5 (the radius).
Now you can easily calculate the base of each triangle. The perimeter is 5 times that.
Thank you, Steve!!
To find the perimeter of a pentagon circumscribed about a circle, we need to know the radius of the circle. In this case, the radius is given as 1.5 units.
First, let's draw a diagram to visualize the problem:
```
B
/ \
/ \
/________\
A B
C D
```
In the diagram, A, B, C, D, and E represent the vertices of the pentagon.
To find the perimeter of the pentagon, we need to find the length of one side and then multiply it by 5 (since a pentagon has 5 equal sides).
Since the circle is circumscribed about the pentagon, the radius of the circle is also the distance from the center (O) of the circle to any vertex of the pentagon (e.g., OA, OB, OC, OD, OE).
Let's calculate the length of one side using basic geometry principles.
The radius (given) is 1.5 units, which means that it is also the distance from the center of the circle to any of its vertices (OA = OB = OC = OD = OE = 1.5 units).
The side length of the pentagon can be found by using the formula:
Side Length = 2 * radius * sin(π/5)
where π is pi (approximately 3.14159).
Substituting the values, we get:
Side Length = 2 * 1.5 * sin(π/5)
Using a calculator, we find:
Side Length ≈ 2 * 1.5 * 0.58779 ≈ 1.76637 units (rounded to 5 decimal places).
Finally, we can find the perimeter by multiplying the side length by 5:
Perimeter = Side Length * 5
Substituting the value we found:
Perimeter = 1.76637 * 5 ≈ 8.83185 units (rounded to 5 decimal places).
Therefore, the perimeter of the pentagon circumscribed about the circle with a radius of 1.5 units is approximately 8.83185 units.
To find the perimeter of a pentagon circumscribed about a circle, we need to use some geometric relationships.
The first thing to note is that in a circumscribed polygon, the vertices of the polygon are tangent to the circle. In the case of a pentagon, each side of the pentagon is tangent to the circle.
Given that the radius of the circle is 1.5 units, we can create a right triangle with one side equal to the radius and the hypotenuse equal to the radius plus the length of one side of the pentagon.
The length of the side of the pentagon can be found using some trigonometry. Since each side of the pentagon is tangent to the circle, we can divide the pentagon into five equal triangles, with the central angle of each triangle being 360°/5 = 72°.
Using the sine function, we can calculate the length of the side of the pentagon:
sin(72°) = opposite / hypotenuse
opposite = 1.5 units * sin(72°)
Now that we have the length of one side of the pentagon, we can multiply it by 5 to find the perimeter:
perimeter = 5 * side length
Let's calculate it step by step:
1. Find the length of one side of the pentagon:
side length = 1.5 units * sin(72°)
2. Calculate the perimeter:
perimeter = 5 * side length
By following these steps, you can find the perimeter of the pentagon circumscribed about the circle with a radius of 1.5 units.