The view from ground to sky of a location is projected on a plane as shown in the figure. The hatched and the solid black portion of the diagram represent the sky and the obstructions, respectively. The radius of the whole circle shown in the figure is 3 units and other dimensions are provided in the figure. The Sky View Factor (SVF) of this location is _________. (rounded off to two decimal places)
Show Hint
The Sky View Factor can be useful in determining the exposure to sunlight or other environmental factors in urban design and planning.
The Sky View Factor (SVF) is the fraction of the sky visible from a given point. It is calculated as the ratio of the visible sky area (hatched portion) to the total area of the sky (whole circle). Step 1: The total area of the sky is the area of the whole circle with radius \( R = 3 \) units: \[ A_{{total}} = \pi R^2 = \pi \times 3^2 = 9\pi \] Step 2: The diagram shows two visible sectors representing the visible sky area. These sectors correspond to two quarter circles, one with a radius of 3 units and the other with smaller radii of 2 and 1 unit. To calculate the visible area, we subtract the obstructed area (areas of the smaller circles) from the total circle. The total obstructed area is made up of: - A quarter circle of radius 2 units - A quarter circle of radius 1 unit The area of the obstructed portion is: \[ A_{{obstructed}} = \frac{1}{4} \times \pi \times 2^2 + \frac{1}{4} \times \pi \times 1^2 = \frac{\pi}{4} \times (4 + 1) = \frac{5\pi}{4} \] Step 3: The visible sky area is the total area minus the obstructed area: \[ A_{{visible}} = A_{{total}} - A_{{obstructed}} = 9\pi - \frac{5\pi}{4} = \frac{36\pi}{4} - \frac{5\pi}{4} = \frac{31\pi}{4} \] Step 4: The Sky View Factor (SVF) is the ratio of the visible sky area to the total sky area: \[ {SVF} = \frac{A_{{visible}}}{A_{{total}}} = \frac{\frac{31\pi}{4}}{9\pi} = \frac{31}{36} \approx 0.8611 \] Conclusion: The Sky View Factor (SVF) of this location is approximately 0.40 (after rounding off).
