Math Problem Statement

Luas lingkaran pada gambar tersebut adalah

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:

  1. Find the area of the entire sector:

    The area of a sector of a circle is given by:

    Area of the sector=θ360×πr2\text{Area of the sector} = \frac{\theta}{360} \times \pi r^2

    where:

    • θ=120\theta = 120^\circ (central angle of the sector)
    • r=6r = 6 cm (radius of the circle)

    Substituting the values:

    Area of the sector=120360×π×62=13×π×36=12πcm2\text{Area of the sector} = \frac{120}{360} \times \pi \times 6^2 = \frac{1}{3} \times \pi \times 36 = 12\pi \, \text{cm}^2
  2. 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:

    Area of the triangle=12r2sin(θ)\text{Area of the triangle} = \frac{1}{2} r^2 \sin(\theta)

    where:

    • r=6r = 6 cm
    • θ=120\theta = 120^\circ

    Substituting the values:

    Area of the triangle=12×62×sin(120)\text{Area of the triangle} = \frac{1}{2} \times 6^2 \times \sin(120^\circ)

    Knowing that sin(120)=32\sin(120^\circ) = \frac{\sqrt{3}}{2}:

    Area of the triangle=12×36×32=93cm2\text{Area of the triangle} = \frac{1}{2} \times 36 \times \frac{\sqrt{3}}{2} = 9\sqrt{3} \, \text{cm}^2
  3. 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:

    Area of the shaded segment=12π93cm2\text{Area of the shaded segment} = 12\pi - 9\sqrt{3} \, \text{cm}^2

Conclusion:

The area of the shaded segment is:

12π93cm212\pi - 9\sqrt{3} \, \text{cm}^2

This corresponds to option D in the given choices.

Would you like more details or have any questions?

Related Questions:

  1. How do you find the area of a segment when the central angle is given in radians?
  2. What is the formula for the area of a triangle when all three sides are known?
  3. Can you explain how to find the length of an arc of a circle?
  4. What happens to the area of a sector if the radius is doubled?
  5. 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!

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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