Calc: Area of Segment = \frac{\theta}{360} \pi r^2 – \frac{1}{2}r^2 \times \frac{\theta}{\frac{\pi}{2}} \frac{21}{11} =
The Correct Answer and Explanation is:
To solve the expression for the area of a segment, we need to carefully understand and simplify the given mathematical formula:Area of Segment=θ360πr2−12r2×θπ2×2111\text{Area of Segment} = \frac{\theta}{360} \pi r^2 – \frac{1}{2} r^2 \times \frac{\theta}{\frac{\pi}{2}} \times \frac{21}{11}Area of Segment=360θπr2−21r2×2πθ×1121
Step 1: Understand the formula
The area of a segment of a circle is given by:Area of Segment=Area of Sector−Area of Triangle\text{Area of Segment} = \text{Area of Sector} – \text{Area of Triangle}Area of Segment=Area of Sector−Area of Triangle
- Area of Sector: θ360πr2\frac{\theta}{360} \pi r^2360θπr2, where θ\thetaθ is in degrees.
- Area of Triangle: often found using trigonometry, or other geometric approximations.
However, the second part of your expression is unusual:12r2×θπ2×2111\frac{1}{2} r^2 \times \frac{\theta}{\frac{\pi}{2}} \times \frac{21}{11}21r2×2πθ×1121
Let’s first simplify this expression step-by-step assuming θ=60∘\theta = 60^\circθ=60∘ and r=7r = 7r=7 (example values — if you have actual values, please provide them).
Step 2: Substitute values (assume r=7r = 7r=7, θ=60\theta = 60θ=60)
First Term:60360π(7)2=16π×49=49π6\frac{60}{360} \pi (7)^2 = \frac{1}{6} \pi \times 49 = \frac{49\pi}{6}36060π(7)2=61π×49=649π
Second Term:12×49×(60π2)×2111\frac{1}{2} \times 49 \times \left( \frac{60}{\frac{\pi}{2}} \right) \times \frac{21}{11}21×49×(2π60)×1121
Simplify inside:60π2=60×2π=120π\frac{60}{\frac{\pi}{2}} = \frac{60 \times 2}{\pi} = \frac{120}{\pi}2π60=π60×2=π120
So the second term becomes:12×49×120π×2111=49×120×212×11π=12348022π=5612.73π≈1786.87\frac{1}{2} \times 49 \times \frac{120}{\pi} \times \frac{21}{11} = \frac{49 \times 120 \times 21}{2 \times 11 \pi} = \frac{123480}{22\pi} = \frac{5612.73}{\pi} \approx 1786.8721×49×π120×1121=2×11π49×120×21=22π123480=π5612.73≈1786.87
First Term (numeric):49π6≈153.9386≈25.656\frac{49\pi}{6} \approx \frac{153.938}{6} \approx 25.656649π≈6153.938≈25.656
Final Answer:
Area of Segment≈25.656−1786.87=−1761.21\text{Area of Segment} \approx 25.656 – 1786.87 = -1761.21Area of Segment≈25.656−1786.87=−1761.21
Explanation (like in textbooks):
To calculate the area of a segment of a circle, we subtract the area of the triangle formed by the chord from the area of the sector defined by the central angle.
- The area of the sector depends on the angle θ\thetaθ and the radius rrr of the circle.
- The area of the triangle is often found using trigonometric identities or geometric relationships. In this case, a custom expression involving θπ2×2111\frac{\theta}{\frac{\pi}{2}} \times \frac{21}{11}2πθ×1121 is used, suggesting a unit conversion or proportion adjustment.
In general, we ensure that angle units are consistent (radians or degrees), and then apply formulas carefully, simplifying each component step by step. The negative result here implies a misinterpretation or incorrect application, as area cannot be negative. Likely, the constants or the form of the second term need to be reviewed or clarified.
