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Chapter 4: Problem 8

Find the area of the sector for the given radius \(r\) and arc length \(s\). \(r=a, s=a\)

Short Answer

Expert verified
The area of the sector is \( \frac{1}{2} a^2 \).

Step by step solution

01

Recall the formula for the area of a sector

The area of a sector of a circle is calculated using the formula: \[ A = \frac{1}{2} r s \] where \( A \) is the area, \( r \) is the radius, and \( s \) is the arc length.
02

Substitute the given values into the formula

Given that the radius \( r = a \) and the arc length \( s = a \), substitute these values into the formula: \[ A = \frac{1}{2} a a \]
03

Simplify the expression

Simplify the expression: \[ A = \frac{1}{2} a^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sector area formula
The area of a sector in a circle can be found using a specific formula. A sector is like a 'slice' of the circle, and the area depends on both the radius of the circle and the length of the arc that forms the sector's edge. The formula to find the area of a sector is: \[ A = \frac{1}{2} r s \]Here:
  • \( A \) is the area of the sector,
  • \( r \) is the radius of the circle,
  • \( s \) is the arc length of the sector.
This formula works because it takes into account the proportional part of the circle's circumference occupied by the arc length. Using this, we can find out how much 'pizza' the slice would occupy if we imagine the circle as a whole pizza.
radius and arc length
The radius and arc length are crucial for calculating the area of a sector. The radius is the distance from the center of the circle to any point on its edge. For example, if you draw a line from the center of a pizza to its edge, that line is the radius.The arc length is a bit different. It is the length of the curved line making up the sector's outer edge. Think of the crust of a pizza slice鈥攊t wraps around the edge of the slice, and its length is the 'arc length'.
  • In our example, the radius was given as \( r = a \).
  • Similarly, the arc length was also given as \( s = a \).
These two values, once plugged into the sector area formula, let us find the area easily.
circle geometry
Understanding circle geometry helps unlock many concepts in mathematics. Every circle has several important parts, including the radius, diameter, circumference, and area. When we talk about sectors (slices of the circle), the key components are:
  • **Radius (r):** the distance from the center to the edge of the circle. All radii in a circle are the same length.
  • **Diameter:** twice the radius, passing through the center from one side of the circle to the other.
  • **Circumference:** the total distance around the circle. It is calculated as \( 2 \pi r \).
  • **Area:** the amount of space inside the circle, given by \( \pi r^2 \).
In sector calculations, we mainly focus on the radius and part of the circumference (the arc length) to find the sector鈥檚 area using the formula \[ A = \frac{1}{2} r s \]In our example, substituting \( r = a \) and \( s = a \) in gave us:\[ A = \frac{1}{2} a a = \frac{1}{2} a^2 \]This simple substitution and understanding of the basic geometry concepts made the calculation straightforward and easy to understand.

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Most popular questions from this chapter

Convert the given angle to degrees. 4 rad

Assume that a particle moves along a circle of radius \(r\) for a period of time \(t\). Given either the arc length \(s\) or the central angle \(\theta\) swept out by the particle, find the linear and angular speed of the particle. \(r=1 \mathrm{~m}, t=1.6\) sec, \(s=3 \mathrm{~m}\)

Convert the given angle to radians. \(4^{\circ}\)

Convert the given angle to degrees. \(\frac{11 \pi}{9} \mathrm{rad}\)

Find the perimeter of a regular dodecagon (i.e. a 12-sided polygon with sides of equal length) inscribed inside a circle of radius \(\frac{1}{2}\). Compare it to the circumference of the circle.
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