Find the area in cm2 of the segment of a circle whose diameter is 20 cm and the central angle measures 700.
To find the area of the segment, we first need to find the length of the chord formed by the central angle of 700° in a circle with a diameter of 20 cm.
The length of the chord can be found using the formula:
Length of chord = 2 * Radius * sin(central angle / 2)
Given that the diameter is 20 cm, the radius is half of the diameter, which is 20 cm / 2 = 10 cm.
Using this information, the length of the chord is:
Length of chord = 2 * 10 cm * sin(700° / 2)
= 2 * 10 cm * sin(350°)
≈ 2 * 10 cm * 0.9397
≈ 18.794 cm
Now, to find the area of the segment, we divide the segment into a triangle and a sector. The area of the segment is the difference between the area of the sector and the area of the triangle.
The sector's area can be found using the formula:
Area of sector = (central angle / 360°) * π * Radius²
Given that the central angle is 700° and the radius is 10 cm, the sector's area is:
Area of sector = (700° / 360°) * π * (10 cm)²
= (7/ 36) * π * 100 cm²
≈ 61.62 cm²
The triangle's area can be found using the formula:
Area of triangle = (1/2) * Length of chord * Radius
Given that the length of the chord is 18.794 cm and the radius is 10 cm, the triangle's area is:
Area of triangle = (1/2) * (18.794 cm) * (10 cm)
≈ 93.97 cm²
Finally, the area of the segment is:
Area of segment = Area of sector - Area of triangle
= 61.62 cm² - 93.97 cm²
≈ -32.35 cm²
However, a negative area is not physically possible for a circle segment. Therefore, there may be an error in the given information or calculations. Please double-check your numbers and confirm the values.
To find the area of the segment of a circle, we can use the formula:
Area = (θ/360) x πr² - (1/2) x r² x sin(θ)
First, we need to find the radius (r) of the circle. Since the diameter is given as 20 cm, the radius is half of the diameter, so:
r = 20/2 = 10 cm
Next, we need to convert the central angle from degrees to radians. Since there are 2π radians in a full circle (360 degrees), we can use the conversion factor:
θ (in radians) = (700 degrees / 360 degrees) x 2π
θ (in radians) = (7/18) x π
θ (in radians) = 7π/18
Now, we can substitute the values into the formula:
Area = (θ/360) x πr² - (1/2) x r² x sin(θ)
Area = (7π/18) / (2π) x π(10)² - (1/2) x (10)² x sin(7π/18)
Simplifying,
Area = (7/18) x π(10)² - (1/2) x (10)² x sin(7π/18)
Area = (7/18) x 100π - 50 x sin(7π/18)
Using a calculator to find the approximate value,
Area ≈ 274.89 cm²
Therefore, the area of the segment of the circle is approximately 274.89 cm².