Given a circle with an 8" radius, find the area of the smaller segment whose chord is 8" long? I don't even understand how to start a question like this?
As usual, draw a diagram. Half the chord is of length 4. Draw the radius through the chord's center. Now you have a right triangle with leg 4 and hypotenuse 8, which subtends an angle x such that
sinx = 1/2
So, x = 30 degrees.
The whole chord thus subtends an angle of 60 degrees.
The area of a circular sector of angle x is 1/2 r^2 x, or in this case, 32π/3
The area of the triangle is 16√3
So, the smaller segment cut by the chord has area 32π/3-16√3
@steve is completely correct!
To find the area of the smaller segment of a circle, you can follow these steps:
Step 1: Draw a circle with an 8" radius.
Step 2: Draw a chord within the circle that is 8" long. The chord should divide the circle into two segments. The smaller segment is the one you're interested in finding the area of.
Step 3: Find the central angle of the smaller segment. To do this, use the formula:
Central angle (in radians) = 2 * arcsin(chord length / (2 * radius))
In this question, the chord length is 8" and the radius is 8". Therefore, the central angle is:
Central angle = 2 * arcsin(8 / (2 * 8))
Step 4: Calculate the area of the smaller segment. To do this, use the formula:
Area of segment = (θ / 2) * r^2 - (1/2) * r^2 * sin(θ)
Where θ is the central angle in radians and r is the radius of the circle.
In this case, the radius is 8" and the central angle (θ) was calculated in Step 3. Therefore, the area of the smaller segment is:
Area of segment = (θ / 2) * 8^2 - (1/2) * 8^2 * sin(θ)
You can plug in the value of θ from Step 3 and calculate the final result to find the area of the smaller segment.
To find the area of the smaller segment of a circle, we need to understand a few concepts first.
A segment of a circle is the region bounded by an arc and a chord. In this case, the chord has a length of 8 inches.
To calculate the area of the smaller segment, follow these steps:
Step 1: Find the angle formed by the chord at the center of the circle.
- Since the chord divides the circle into two equal arcs, the angle formed at the center by the chord is 180 degrees.
Step 2: Find the value of the smaller angle formed by the chord at the center of the circle.
- To do this, divide the angle by 2: 180 degrees / 2 = 90 degrees.
Step 3: Find the length of the arc intercepted by the smaller angle.
- Since the smaller angle is 90 degrees and the radius of the circle is 8 inches, you can use the formula: arc length = (angle/360) * (2 * π * radius).
- Plug in the values: arc length = (90/360) * (2 * π * 8).
Step 4: Find the height of the segment.
- The height of the segment is the distance from the center of the circle to the midpoint of the chord.
- It can be calculated using the Pythagorean theorem: height = √(radius^2 - (chord length/2)^2).
- Plug in the values: height = √(8^2 - (8/2)^2).
Step 5: Calculate the area of the smaller segment.
- The area of the segment is equal to the area of the sector of the circle minus the area of the triangle formed by the chord and the height.
- The area of the sector is given by: (angle/360) * (π * radius^2).
- The area of the triangle is given by: (1/2) * (chord length) * (height).
- Plug in the values: area of segment = (90/360) * (π * 8^2) - (1/2) * 8 * √(8^2 - (8/2)^2).
By following these steps, you should be able to find the area of the smaller segment in square units.