Problem 37 - Basic Technical Mathematics with Calculus

Question

In a circle, a chord connects the ends of two perpendicular radii of 6.00 in. What is the area of the minor segment?

Step-by-Step Solution

Answer

The area of the minor segment is \(9\pi - 18\) square inches.

Step 1: Identify Key Components

First, draw the circle with a center O and two perpendicular radii, OA and OB, each 6 inches long. The ends of the radii form points A and B, and the chord AB runs between these two points.

Step 2: Calculate the Length of the Chord

Since OA and OB form a right triangle with hypotenuse AB, use the Pythagorean theorem to determine the length of AB: \[AB = \sqrt{OA^2 + OB^2} = \sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2} \text{ inches}.\]

Step 3: Determine the Central Angle

The angle AOB is 90 degrees because OA and OB are perpendicular. To work with radians, convert this angle: \[90° = \frac{\pi}{2} \text{ radians}.\]

Step 4: Find the Area of the Sector

The area of the sector AOB can be found using the formula: \[\text{Area of Sector AOB} = \frac{1}{2} \times 6^2 \times \frac{\pi}{2} = \frac{36\pi}{4} = 9\pi \text{ square inches}.\]

Step 5: Calculate the Triangle Area

Since triangle OAB is a right triangle, use the formula for the area of a right triangle:\[\text{Area of } \triangle OAB = \frac{1}{2} \times 6 \times 6 = 18 \text{ square inches}.\]

Step 6: Determine the Area of the Segment

The segment is the portion of the sector OAB that does not include the triangle OAB. Calculate the segment area:\[\text{Area of Segment} = 9\pi - 18 \text{ square inches}.\]

Key Concepts

Understanding Chord Length in a Circle

A chord in a circle is a straight line connecting two distinct points on the circle itself. In our exercise, this line is represented by the distance between points A and B, known as the chord AB. To find the length of this chord when two radii meet at a right angle, we employ the Pythagorean theorem. This is because the structure formed is a right triangle, with the radii OA and OB as its legs. In this case, the lengths of these radii are each 6 inches. Therefore, applying the theorem, the chord AB is the hypotenuse of the right triangle OAB.
To calculate the length, the formula is:

• First, square the lengths of the radii: \(OA^2 = 6^2 = 36\) and \(OB^2 = 6^2 = 36\).
• Add these squares: \(36 + 36 = 72\).
• Finally, take the square root: \(AB = \sqrt{72} = 6\sqrt{2}\) inches.

So, the length of the chord AB is approximately 8.49 inches. In circle geometry, finding the chord length is crucial for further calculations, such as determining areas or other lengths within the circle.

The Role of Right Triangles in Circle Geometry

Right triangles are pivotal in circle geometry, especially when dealing with perpendicular radii. In geometry, right triangles help identify relationships and calculations that aren't immediately obvious. Any two radii meeting at the center of a circle form two lines that are perpendicular if the central angle between them is 90 degrees. This angle makes the triangle a special type of triangle known as a right triangle.
Right triangles allow for the application of the Pythagorean theorem, which is invaluable in calculating the length of unknown sides. In our exercise, the perpendicular radii OA and OB form a right triangle OAB with the chord AB as the hypotenuse.
Understanding this concept helps in visualizing the geometric properties of the setup, and it lays the groundwork for computing areas, such as sector and segment areas, as well as facilitating transitions into other geometric calculations, like finding the central angle and areas of segments or sectors.

Exploring the Central Angle and its Importance

The central angle of a circle is the angle formed by two radii from the center of the circle to two points on its circumference. For instance, the central angle in our circle is the angle \(\angle AOB\), formed by radii \(OA\) and \(OB\), which measure 90 degrees. When expressed in radians, this angle becomes \(\frac{\pi}{2}\).
Central angles are critical in circle geometry because:

• They determine the size of the arc and the corresponding sector.
• They are instrumental in calculating the areas of sectors and segments.
• They help in understanding the proportional relationship within the circle's geometry.

To work with central angles in radians allows for an easier integration with other mathematical formulas and provides a uniform measurement system that is universally accepted in advanced mathematics.

Calculating Sector Area in a Circle

The sector of a circle is a "pie-shaped" part of the circle, defined by two radii and the arc between them. The area of a sector is determined by the circle's radius and the central angle. This area can be thought of as the fraction of the circle occupied by the sector.
In our exercise, the sector AOB has:

• A radius of 6 inches.
• A central angle of \(\frac{\pi}{2}\) radians (from 90 degrees).

The formula to find the area of the sector is \[ \text{Area of Sector} = \frac{1}{2} imes r^2 imes \text{Central Angle} \]Plugging in the values:

• \(r = 6\) inches.
• The area of the sector \(= \frac{1}{2} \times 6^2 \times \frac{\pi}{2} = 9\pi\) square inches.

This calculation is essential for understanding how much of the circle is occupied by the sector and is also crucial when determining the size of any segments within the sector itself.