Math Problem Statement

Find the area

Find the area of the sector in the circle with a radius of 12 units and an angle of 0.5 radians.

Solution

To find the area of the sector shown in the image, we use the formula for the area of a sector of a circle:

A=12r2θA = \frac{1}{2} r^2 \theta

Where:

  • rr is the radius of the circle,
  • θ\theta is the angle in radians.

From the image:

  • The radius r=12r = 12,
  • The angle θ=0.5rad\theta = 0.5 \, \text{rad}.

Now, let's compute the area AA.

A=12×122×0.5A = \frac{1}{2} \times 12^2 \times 0.5

Let me calculate that for you.The area of the sector is 36square units36 \, \text{square units}.

Would you like further details or have any other questions?

Here are 5 related questions you might find interesting:

  1. How do you calculate the area of a full circle?
  2. What is the circumference of a circle with radius 12 units?
  3. How would the area change if the angle were in degrees instead of radians?
  4. What is the formula for the length of an arc of a sector?
  5. How do you calculate the perimeter of a sector, including the arc length?

Tip: Always ensure that the angle in your formula is in radians when calculating areas or lengths related to circular sectors.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circle
Radians
Sector Area

Formulas

A = 1/2 * r^2 * θ

Theorems

Area of a sector of a circle

Suitable Grade Level

Grades 8-10