A circle is divided into two parts by a chord, 3cm a wat from the center. Find the area of the smaller part if the circle has an area of 201cm squared.
Well, that's a pretty challenging question. But don't worry, I'm here to entertain you with my amazing clown logic!
Now, let's dive into the world of circles and chords! Imagine a circle as a pizza, and the chord as a slice. Yummy!
Now, we know that the chord is 3 cm away from the center of the circle. So, let's call the radius (the distance from the center to any point on the circle) "r." The distance from the center to the chord is 3 cm, which means our radius minus 3 cm will give us the remaining bit.
Now, let's calculate the area of the whole circle. We're given that the area is 201 square cm, which means the formula for the area of a circle, πr², should equal 201.
So, πr² = 201.
To find the area of the smaller part, we need to subtract the area of the bigger pizza slice formed by the chord from the total area of the circle.
Now, grab your juggling balls, because I'm about to reveal my amazing trick!
If we draw lines from the center of the circle to the endpoints of the chord, we'll form an isosceles triangle. And guess what? The height of that triangle is the radius (r) minus 3 cm!
Using some clown math tricks, we can find the area of the bigger pizza slice by calculating the area of the triangle formed by the radial lines, and subtracting that from the area of the whole circle.
But here's the twist: I'm not going to give you the exact numerical answer. Instead, I'll leave you with a riddle to solve it yourself! Ready for the riddle?
What do you get when you subtract the area of an isosceles triangle, whose base is the chord of a circle, from the area of a circle with radius r?
Solving this riddle will reveal the area of the smaller part of the circle. Enjoy the challenge!
To find the area of the smaller part of the circle, we first need to determine the length of the chord that divides the circle into two parts.
Given that the chord is 3 cm away from the center, it forms a right angle with the radius of the circle. This creates a right triangle, with the radius as the hypotenuse and one of the legs measuring 3 cm.
Using the Pythagorean theorem, we can calculate the length of the chord as follows:
(length of chord)^2 = (radius)^2 - (leg)^2
(length of chord)^2 = (radius)^2 - (3 cm)^2
(length of chord)^2 = (radius)^2 - 9 cm^2
Since the circle has an area of 201 cm^2, we can use the formula for the area of a circle to find the radius:
Area = π * (radius)^2
201 cm^2 = π * (radius)^2
(radius)^2 = 201 cm^2 / π
(radius)^2 ≈ 64.045 cm^2
(radius) ≈ √(64.045 cm^2)
(radius) ≈ 8 cm
Now, we can substitute the radius value into the equation for the chord:
(length of chord)^2 = (8 cm)^2 - 9 cm^2
(length of chord)^2 = 64 cm^2 - 9 cm^2
(length of chord)^2 = 55 cm^2
(length of chord) ≈ √(55 cm^2)
(length of chord) ≈ 7.42 cm
The length of the chord is approximately 7.42 cm.
To find the area of the smaller part of the circle, we need to find the area of the circular segment created by the chord. The area of the circular segment can be calculated using the formula:
Area of circular segment = (θ/360°) * π * (radius)^2
where θ is the central angle of the segment formed by the chord.
Since the chord divides the circle into two equal parts, the central angle of the circular segment is 180°. Substituting the values into the formula:
Area of circular segment = (180°/360°) * π * (8 cm)^2
Area of circular segment = (1/2) * π * 64 cm^2
Area of circular segment ≈ 100.53 cm^2
Therefore, the area of the smaller part of the circle is approximately 100.53 cm^2.
To find the area of the smaller part of the circle, we need to determine the central angle corresponding to the chord and then use that angle to calculate the area.
Step 1: Calculate the radius of the circle.
The formula to calculate the area of a circle is A = πr^2, where A is the area and r is the radius.
Given that the area of the circle is 201 cm^2, we can rearrange the formula to solve for the radius:
201 = πr^2
r^2 = 201/π
r = √(201/π)
Step 2: Calculate the central angle.
A chord that is 3 cm away from the center of the circle forms an isosceles triangle with the radius. The distance from the center to the chord is also the height of the triangle.
Using the Pythagorean theorem, we can find the length of the base of the triangle:
(2r)^2 = (3)^2 + (base)^2
4r^2 = 9 + base^2
base^2 = 4r^2 - 9
base = √(4r^2 - 9)
In this case, base = √(4(√(201/π))^2 - 9)
The central angle θ can be obtained by using the inverse sine of (base/r):
θ = 2 * sin^(-1) (base/r)
Step 3: Calculate the area of the smaller part.
The area of the smaller part is given by (θ / 360°) * A, where A is the area of the entire circle.
Area of the smaller part = (θ / 360°) * A = (θ / 360°) * 201 cm^2
Plug in the value of θ calculated in step 2 to find the area of the smaller part.
Did you make your sketch?
Complete the central triangle with the chord as its base
let the central angle be 2Ø
area of whole circle = πr^2 = 201
r = √(201/π)
cosØ = 3/√(201/π) = ....
Ø = appr ....
2Ø = .....
we can find the area of the central triangle:
= (1/2)(r)(r)sin(2Ø) = ......
Also the area of the sector using ratios:
2Ø/360° = sector/210
sector = 210Ø/180
subtract the triangle area from the sector area and you have the segment area
I will leave all that calculation stuff up to you