A 7.5 minute quadrangle covers land at a scale of 1:24000 and spans 7.5 minutes of arc in each direction. Approximately how many square miles does it cover?

A 7.5 minute quadrangle covers land at a scale of 1:24000 and spans 7.5 minutes of arc in each direction. Approximately how many square miles does it cover?

• A. 49-70 square miles
• B. 10-20 square miles
• C. 100-120 square miles
• D. 200-250 square miles

Answer: (A)

Think of the 7.5-minute quadrangle as a small rectangle on the Earth’s surface. The north–south side spans 7.5 minutes of latitude, which is 0.125 degrees. One degree of latitude is about 69 miles, so the north–south distance is around 8.6 miles. The east–west side spans 7.5 minutes of longitude; the distance per degree of longitude is 69 miles times the cosine of the latitude, so the east–west distance is roughly 8.625 × cos(latitude) miles. At typical mid-latitudes, cos(latitude) is about 0.75–0.80, giving an east–west distance of roughly 6.5–6.9 miles. Multiplying the two sides gives an area around 50–60 square miles. That falls in the approximate range of about 49–70 square miles, which is why that option is the best estimate. The map’s scale doesn’t change the ground area, it just relates map measurements to real distances.

Think of the 7.5-minute quadrangle as a small rectangle on the Earth’s surface. The north–south side spans 7.5 minutes of latitude, which is 0.125 degrees. One degree of latitude is about 69 miles, so the north–south distance is around 8.6 miles. The east–west side spans 7.5 minutes of longitude; the distance per degree of longitude is 69 miles times the cosine of the latitude, so the east–west distance is roughly 8.625 × cos(latitude) miles. At typical mid-latitudes, cos(latitude) is about 0.75–0.80, giving an east–west distance of roughly 6.5–6.9 miles. Multiplying the two sides gives an area around 50–60 square miles. That falls in the approximate range of about 49–70 square miles, which is why that option is the best estimate. The map’s scale doesn’t change the ground area, it just relates map measurements to real distances.