Question

An arc of a circle, radius \(5 \mathrm{~cm}\), subtends an angle of \(\frac{3 \pi}{4}\) radians at the centre. Calculate the length of the arc.

Short answer

Answer: The length of the arc is approximately 11.78 cm.

Step-by-step solution

Write down the given values

We are given the radius r = 5 cm and the angle θ = (3π/4) radians.

Use the arc length formula

The formula for arc length is S = r × θ. Plug in the given values: S = (5 cm) × (3π/4)

Solve for S (arc length)

Multiply the values we plugged in: S = 15π/4 cm

Final Answer

The length of the arc is 15π/4 cm or approximately 11.78 cm.

Key concepts

Circle Geometry

Circle geometry is a branch of geometry focused on the properties and dimensions of a circle. A circle is a simple, yet profoundly important geometric shape. It consists of all points in a plane that are at a constant distance, known as the radius, from a central point. The central point is called the "center" while the boundary is referred to as the "circumference." Points along the circumference create angles at the center when lines are drawn from the center out to the circumference. Understanding the circle's components is crucial, such as:

• Radius (r): The constant distance from the center to any point on the circle.
• Diameter (d): Twice the radius, or a line that passes through the center connecting two points on the circumference. It is calculated as \( d = 2r \).
• Circumference (C): The total distance around the circle, calculated by \( C = 2\pi r \).

Circle geometry provides the foundation for understanding various properties and theorems related to arcs, chords, sectors, and other segments within a circle. These key concepts form an essential backdrop when calculating specific parts of a circle, such as its arc length.

Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians provide a natural and mathematical consistency by linking the angle with the arc of the circle. An angle in radians is determined by the formula:\[ \theta (radians) = \frac{\text{Arc Length}}{\text{Radius}} \]This makes radians incredibly useful in various mathematical calculations and real-world applications, including physics and engineering. To give some perspective:

• One full rotation around a circle (360 degrees) is equal to \( 2\pi \) radians.
• Half a rotation (180 degrees) is \( \pi \) radians.
• Quarter of a circle (90 degrees) equals \( \pi/2 \) radians.

Understanding radian measure is crucial when working with circular shapes and is particularly useful when calculating arc lengths, as demonstrated in our problem.

Arc Formula

The arc formula is a pivotal tool in circle geometry, allowing us to calculate the length of an arc. The arc length can be thought of as a segment of the circumference of a circle. Calculating it requires knowing both the radius of the circle and the central angle subtended by the arc in radians. The arc length formula is given by:\[ S = r \times \theta \]Where:

• \( S \) is the arc length.
• \( r \) is the radius of the circle.
• \( \theta \) is the central angle in radians.

Applying this formula is straightforward once the angle is known in radians. For example, in our exercise where the radius \( r = 5 \) cm and angle \( \theta = \frac{3\pi}{4} \), substituting these values into the formula gives us: \[ S = 5 \times \frac{3\pi}{4} = \frac{15\pi}{4} \text{ cm} \]This calculation reflects the arc length, which represents a specific portion of the whole circle's circumference. Understanding how to apply this formula effectively will enable accurate computations of diverse arc lengths in various problems.