Math Problem Statement
Luas lingkaran pada gambar tersebut adalah
Solution
The problem in the image involves finding the area of a shaded segment in a circle. Let's go through the information presented in the image to solve it.
The image shows a circle with a sector, and the central angle of the sector is labeled as 120°. The radius of the circle is given as 6 cm.
To find the area of the shaded segment, we can use the formula for the area of a sector and the formula for the area of a triangle formed by the radii and the chord.
Step-by-step Solution:
-
Find the area of the entire sector:
The area of a sector of a circle is given by:
where:
- (central angle of the sector)
- cm (radius of the circle)
Substituting the values:
-
Find the area of the triangle formed by the radii and the chord:
The triangle formed within the sector is an equilateral triangle (since the central angle is 120° and the two radii are equal).
The area of the triangle can be found using the formula:
where:
- cm
Substituting the values:
Knowing that :
-
Find the area of the shaded segment:
The area of the shaded segment is the area of the sector minus the area of the triangle:
Conclusion:
The area of the shaded segment is:
This corresponds to option D in the given choices.
Would you like more details or have any questions?
Related Questions:
- How do you find the area of a segment when the central angle is given in radians?
- What is the formula for the area of a triangle when all three sides are known?
- Can you explain how to find the length of an arc of a circle?
- What happens to the area of a sector if the radius is doubled?
- How do you calculate the perimeter of a sector in a circle?
Tip:
Remember that for sectors, both the radius and the central angle are crucial for calculations, so make sure to always check if the angle is given in degrees or radians!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Geometry
Circle Area
Sector Area
Triangle Area
Trigonometry
Formulas
Area of a sector: (θ/360) × πr^2
Area of a triangle: (1/2) × r^2 × sin(θ)
Theorems
Trigonometric properties of sine
Sector and segment of a circle
Suitable Grade Level
Grades 9-11
Related Recommendation
Calculate the Shaded Area of a Sector with θ = 30° and Radius = 6 cm
Finding the Area of the Shaded Region in a Circle with a Sector and Chord
Calculate Area of Shaded Region in Circle with 102° Central Angle and 9 cm Radius
Calculate the Area of a Shaded Region in a Circle with Radius 5
Calculate Area of Shaded Segment in Circle with Radius 8 cm and Chord 14 cm