What formula gives the length of an arc and what unit must the angle use?
Arc length S = r·θ, where r is the radius and θ must be in radians.
Video Summary
Arc Length and Area of a Sector
Main takeaways
01
Arc length S is given by S = r·θ when θ is in radians; radians measure arc length relative to the radius.
02
Convert degrees to radians by multiplying by π/180 before using S = rθ or A = 1/2 r²θ.
03
Example: arc with r = 9.5 cm and θ = 120° → θ = 2π/3, exact S = 19π/3 cm ≈ 19.9 cm.
04
Area of a sector A = 1/2 · r² · θ; example: r = 450 m, θ = 240° → θ = 4π/3, exact A = 135,000π m² ≈ 424,115 m².
05
Compare arc lengths by computing S for each radius with the same θ and subtract; e.g., NCAA vs international free-throw arcs differ by ≈0.298 ft (≈3.581 in).
Key moments
Questions answered
How do you convert 120 degrees to radians and what is the exact arc length for r = 9.5 cm?
Multiply by π/180: 120° = 2π/3 radians. Exact arc length S = 9.5 × 2π/3 = 19π/3 cm (≈19.9 cm).
How do you compute the area of a sector and what is the area for r = 450 m and θ = 240°?
Sector area A = 1/2 · r² · θ with θ in radians. 240° = 4π/3, so A = 1/2 · 450² · 4π/3 = 135,000π m² (≈424,115 m²).
How was the difference between two half-circle arcs on basketball courts calculated and what is the result?
Compute S = r·π for each radius, subtract the smaller from the larger. The difference is ≈0.298 feet, which is about 3.581 inches.
Why is it important to use radians in the formulas S = rθ and A = 1/2 r²θ?
Those formulas derive from the fraction of a full circle using θ/2π; when θ is in radians π cancels cleanly, giving the simple forms S = rθ and A = 1/2 r²θ.
Arc Length and Its Calculation 00:12
"The length of an arc of a circle depends on both the angle of rotation and the radius length of the circle."
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The length of an arc in a circle is influenced by two main factors: the angle of rotation and the radius of the circle.
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A radian is defined as the angle that subtends an arc whose length is equal to the radius of the circle.
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For example, a half rotation corresponds to π radians, which is slightly more than three times the radius length of the circle.
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If the radius of a circle is 4 centimeters, the length of the half-circle arc is expressed as 4/π centimeters.
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Additionally, dividing the arc length by the radius gives the angle measure in radians, confirming that the angle would be π radians in this case.
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The general formula for arc length (S) can be derived as S = radius × θ, where θ must be expressed in radians.
Finding Arc Length Example 01:30
"To find the length of the arc intercepted by a central angle of 120 degrees, we must first convert this angle into radians."
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In a specific example, to find the arc length when the radius is 9.5 centimeters and the central angle is 120 degrees, we first convert degrees to radians.
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The conversion involves multiplying by π/180; thus, 120 degrees converts to 2π/3 radians.
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The arc length can then be calculated using the formula ( S = \text{radius} \times \theta ), resulting in ( S = 9.5 \times \frac{2\pi}{3} ).
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The decimal approximation of this value is approximately 19.9 centimeters, but the exact arc length in terms of π is ( \frac{19\pi}{3} ) centimeters.
Comparing Two Arc Lengths on a Basketball Court 03:26
"To compare the lengths of two arcs, we need to find the difference in their dimensions."
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In examining the half-circle arcs above the free throw lines of NCAA and international basketball courts, we denote the outer arc length as A and the inner arc length as B.
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For the outer arc, with a radius of 6 feet and an angle of π radians, the arc length A can be calculated leaving the answer in terms of π.
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The smaller inner arc has a radius of 11.81 feet, thus for S-B, the angle remains π.
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The difference in arc lengths is computed, resulting in approximately 0.298 feet. This difference can be converted to inches, yielding about 3.581 inches.
Area of a Sector 05:40
"The area of a sector is derived from the fraction of the circle defined by the central angle in radians."
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A sector of a circle represents a portion of the circle's interior, defined by a central angle.
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When θ (the angle in radians) is employed, the area of the sector can be articulated as ( \frac{1}{2} \times R^2 \times \theta ), simplifying to ( \frac{1}{2} R^2 \theta ) after canceling out π in the calculation.
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In a practical example, irrigation through a center pivot method defines circular fields, and if the irrigation pipe is 450 meters, we can find the irrigated area after a rotation of 240 degrees.
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The area calculation requires converting 240 degrees into radians, which gives ( \frac{4π}{3} ).
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Finally, the exact area formula provides ( 135,000π ) square meters, which converts to an approximate decimal area of 424,115 square meters.