Q. The chord has a length of 100 and the area enclosed by the chord has an area of 400. How do you find the radius of the circle and what is it?
A. The formula for the area, A, of a chord is
A = R² arccos(r / R) - r â(R² - r²)
where R is the radius of the circle.
r is the distance between the chord and
the center of the circle. In other words
â(R² - r²) = w/2 where w is the length of
the chord.
You were given A = 400 and w = 100. So
400 = R² arccos(r/R) - 50r
R² arccos(r/R) = 400 + 50r
arccos(r/R) = (400 + 50r)/R²
cos( (400 + 50r)/R²) = r/R
cos²( (400 + 50r)/R²) = r²/R²
where â(R² - r²) = 50 implies
r² = R² - 2500
So we need to solve the equations
R² = r² + 2500
cos²( (400 + 50r)/R²) = r²/R²
OR
cos²( (400 + 50r)/(2500 + r²)) = r²/(2500 + r²)
sin²( (400 + 50r)/(2500 + r²)) = 2500/(2500 + r²)
Let r = 50x, then
sin²( (x + 0.16)/(x² + 1)) = 1/(x² + 1)
Wolfram alpha says that x = 4.11874506121247
So
r = 205.9372530606235
R = 211.92015524285386649
check:
R² arccos(r / R) - r â(R² - r²) = 399.99999999999969
So the radius of your circle is about 211.92015524285386649
A = R² arccos(r / R) - r â(R² - r²)
where R is the radius of the circle.
r is the distance between the chord and
the center of the circle. In other words
â(R² - r²) = w/2 where w is the length of
the chord.
You were given A = 400 and w = 100. So
400 = R² arccos(r/R) - 50r
R² arccos(r/R) = 400 + 50r
arccos(r/R) = (400 + 50r)/R²
cos( (400 + 50r)/R²) = r/R
cos²( (400 + 50r)/R²) = r²/R²
where â(R² - r²) = 50 implies
r² = R² - 2500
So we need to solve the equations
R² = r² + 2500
cos²( (400 + 50r)/R²) = r²/R²
OR
cos²( (400 + 50r)/(2500 + r²)) = r²/(2500 + r²)
sin²( (400 + 50r)/(2500 + r²)) = 2500/(2500 + r²)
Let r = 50x, then
sin²( (x + 0.16)/(x² + 1)) = 1/(x² + 1)
Wolfram alpha says that x = 4.11874506121247
So
r = 205.9372530606235
R = 211.92015524285386649
check:
R² arccos(r / R) - r â(R² - r²) = 399.99999999999969
So the radius of your circle is about 211.92015524285386649
Find the area of a segment formed by a chord 8 inches long in a circle with radius of 8 inches?
Q. To answer this question, give the correct formula on how to solve it.
A. You will need several formulas to solve this:
area of the segment = area of the sector - area of the triangle bound by the chord and the radii
* you have to solve for the area of the sector and the area of the triangle
area of the sector = pi * r^2 * (angle / 360)
* You need to determine the angle. You can calculate this via trigonometry (i.e. cosine law). However, take note that the triangle bound by the chord and the radii is an equilateral triangle. Hence, the angle in this problem is 60 degrees.
area of the triangle = 1/2 bh
* b = the length of the chord; h = the height of the triangle, calculate this using trigonometry
You now have everything you need to solve the area of the segment.
area of the segment = area of the sector - area of the triangle bound by the chord and the radii
* you have to solve for the area of the sector and the area of the triangle
area of the sector = pi * r^2 * (angle / 360)
* You need to determine the angle. You can calculate this via trigonometry (i.e. cosine law). However, take note that the triangle bound by the chord and the radii is an equilateral triangle. Hence, the angle in this problem is 60 degrees.
area of the triangle = 1/2 bh
* b = the length of the chord; h = the height of the triangle, calculate this using trigonometry
You now have everything you need to solve the area of the segment.
What is the formula for finding a chord in a circle?
Q. also, if you know the radius, if there is a circle that has radius of 10, and a chord intercepts the radius, If PQ = 10 and PC = 6, what is AB?
if the chord intercepts at c, the radius is from PCQ
if the chord intercepts at c, the radius is from PCQ
A. the chord segments have the following lengths: A= 6, C=3, D=4. Use the theorem for the product of chord segments to find the value of D.
http://www.mathwarehouse.com/geometry/circle/product-segments-chords.php
http://www.mathwarehouse.com/geometry/circle/product-segments-chords.php
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