Hubungan antara Sudut Pusat, Panjang Busur dan Luas Juring

Summary

This video tutorial explores the relationship between central angles, arc lengths, and sectors of a circle. It explains how to calculate the arc length and area of a sector using formulas based on the central angle and radius. Through practical examples, viewers learn to derive these values and understand their interconnections, emphasizing the importance of correct unit and variable handling. The session also includes comparisons of angles and their corresponding areas, providing a comprehensive understanding of circular geometry concepts.

Takeaways

• 😀 The relationship between central angles, arc lengths, and sector areas is essential for understanding circles.
• 😀 The formula for arc length (AB) is derived from the proportion of the central angle to 360 degrees multiplied by the circle's circumference.
• 😀 The formula for the area of a sector (OAB) involves the central angle, with the sector area proportional to the entire area of the circle.
• 😀 The area of a segment can be calculated by subtracting the area of the triangle from the area of the sector.
• 😀 An example with a central angle of 90° and a radius of 14 cm illustrates how to calculate arc length and sector area.
• 😀 For a 90° angle, the arc length is 22 cm and the area of the sector is 154 cm².
• 😀 The formula for arc length can also be simplified using fractions for easier calculation.
• 😀 Understanding the proportional relationships between angles and corresponding lengths or areas is crucial for problem-solving.
• 😀 When given two angles and one arc length, you can find the length of another arc by setting up a proportion.
• 😀 The final example shows how to calculate the area of a sector using known areas and angles, concluding with a sector area of 60 cm².

Q & A

What is the relationship between the central angle and the arc length in a circle?

The arc length is proportional to the central angle. Specifically, the length of the arc is calculated using the formula: arc length = (central angle in degrees / 360) × circumference of the circle.

How is the area of a sector related to the central angle?

The area of a sector (or juring) is also proportional to the central angle. It can be calculated with the formula: area = (central angle in degrees / 360) × area of the entire circle.

What are the formulas for calculating arc length and area of a sector?

The formulas are: Arc Length = (central angle in degrees / 360) × 2πr and Area of Sector = (central angle in degrees / 360) × πr².

In the example provided, how is the arc length for a 90° angle with a radius of 14 cm calculated?

Using the formula for arc length: Arc Length = (90° / 360°) × 2π(14). This simplifies to 22 cm after calculation.

What is the method for calculating the area of a sector in the example?

The area of the sector is calculated using the formula: Area = (90° / 360°) × π(14)², which results in 154 cm².

How do you find the area of a segment in the example?

The area of the segment is found by subtracting the area of triangle AOB from the area of the sector. The final result for the segment area is 56 cm².

What relationship is established between two central angles in the discussion?

The ratio of the two central angles is equal to the ratio of their corresponding arc lengths and the areas of their sectors.

How is the arc length CD calculated in the second example?

Using the proportion of angles and arc lengths, the length of arc CD is derived from the equation: (50° / 80°) = (40 cm / arc length CD).

What does the calculation of area OAB in relation to area OBC demonstrate?

It shows that if the central angle OAB is smaller than BOC, then the area of sector OAB will also be proportionally smaller than that of sector OBC.

What should you keep in mind when comparing angles, arc lengths, and areas?

Ensure that the ratios maintain the relationship; if one parameter is known, others can be calculated based on the established proportions.