A 4cm chord subtands a central angle of 60 degree.The radial segment of this circle is
Your question is not clear.
Do you want the area of the segment or the length of the
arec of that segment.
If arclength:
construct the triangle by using the radii
Clearly you have an equilateral triangle , all sides being 4 cm
the circumference of the whole circle is 2πr = 8π m
so the arc length of the segment is (60/360)(8π) cm = ...
if area of arc:
area of whole circle = πr^2 = 16π cm^2
area of sector = (1/6)(16π) cm^2 = ....
area of the equilateral triangle = (1/2)(4)(4)sin60 = ....
area of segment = area of sector - area of triangle
To find the length of the radial segment, we first need to determine the radius of the circle.
We know that the chord in question subtends a central angle of 60 degrees. The chord divides the circle into two equal parts, and the central angle divides the circle into two equal arcs.
So, the arc subtended by the 60-degree central angle is also 60 degrees.
The circumference of a circle is given by the formula C = 2πr, where C represents the circumference and r represents the radius.
Since the arc subtended by the 60-degree central angle is equal to 60 degrees, it means that the length of this arc is 1/6th of the entire circumference.
Thus, the length of the arc is (1/6) * (2πr) = (πr) / 3.
We are given that the length of the chord is 4 cm. Using some basic trigonometry, we can relate the length of the chord, the radius, and the central angle.
In a circle, the radius bisects the chord at a right angle. This creates a right triangle, where the radius is the hypotenuse, the chord is the base, and half the chord length is the adjacent side.
Using the trigonometric relationship cosine (cos), we can write:
cos(30 degrees) = adjacent side / hypotenuse
cos(30 degrees) = (1/2) * chord length / radius
Plugging in the values we know, we can solve for the radius:
cos(30 degrees) = (1/2) * 4 / radius
cos(30 degrees) = 2 / radius
Since cos(30 degrees) = √3 / 2, we have:
√3 / 2 = 2 / radius
Cross-multiplying, we get:
radius = 2 * 2 / √3
radius = 4 / √3
To simplify this expression, we can rationalize the denominator by multiplying both the numerator and denominator by √3:
radius = (4 / √3) * (√3 / √3)
radius = 4√3 / 3
Therefore, the radial segment of this circle is 4√3 / 3 cm.
To find the length of the radial segment of a circle, we can use the relationship between the central angle and the length of the chord.
Given:
- Chord length = 4 cm
- Central angle = 60 degrees
1. Firstly, let's find the length of the radius.
As we know, the radius intersects the chord at a right angle, dividing the chord into two equal parts.
We can draw an equilateral triangle using the chord, where each side is equal to the chord length.
2. In an equilateral triangle, the central angle of each vertex is 60 degrees since all angles are equal.
So, if we extend one side of the triangle to the center of the circle, it will create a triangle with a central angle of 60 degrees.
3. Now, we have a right-angled triangle with one angle of 60 degrees and the opposite side equal to half the length of the chord (2 cm).
Using trigonometry, we can find the length of the radius.
Let's use the sine function, which relates the opposite side to the hypotenuse:
sine(angle) = opposite/hypotenuse
4. Plugging in the values:
sine(60 degrees) = 2 cm / radius
5. Rearranging the equation to solve for the radius:
radius = 2 cm / sine(60 degrees)
6. Evaluating the sine of 60 degrees:
sine(60 degrees) = √3 / 2
7. Substituting the value back into the equation:
radius = 2 cm / (√3/2) = 4 cm / √3
Thus, the length of the radial segment of this circle is 4 cm / √3 cm, which is approximately equal to 2.31 cm (rounded to two decimal places).