set up proportion: C 360 -- = --- d angle (formed between two points on Earth) where: C = circumference, Earth d = distance, between two points on Earth 360 = number of degrees in a circle angle = angle formed between two points on the surface of Earth and the center of Earth Note: "angle" may be found by finding the difference in latitude between your location and the latitude of the subsolar point refer to either... NASS website, "Current Solar Data" (realtime): http://www.sundials.org/ or... Table of the Declination of the Sun http://www.wsanford.com/~wsanford/exo/sundials/DEC_Sun.html APR 22 = +12deg00' solve for C: C = 360 x d / angle == Date: Tue, 31 May 2005 13:55:58 -0400 (EDT) Subject: "Girth Day" - Eratosthenes' Measurement of Earth's Circumference Like "Earth Day." 22 APR 2005 was the 35th anniversary of Earth Day. http://www.earthday.net/ -- Merriam-Webster Online Main Entry: 1girth Pronunciation: 'g&rth Function: noun Etymology: Middle English, from Old Norse gjorth; akin to Old English gyrdan to gird 1 : a band or strap that encircles the body of an animal to fasten something (as a saddle) on its back 2 a : a measure around a body b : SIZE, DIMENSIONS -- George F. Cram 12" Physical Terrestrial Political Globe [printed on globe] Diameter: 12" = 30.48 cm [Circumference: 95.7072 cm] Natural Scale: 1:41,800,000 1 cm = 41.8 km <-- this scale CANNOT be correct! (double-check) True Values: 40,008 km = polar circumference 40,076 km = equatorial circumference -- A regulation NBA basketball is 9" in diameter, ... 28.27" in circumference. Google Search: D = 9 inches = ? cm D = 9 inches = 22.86 centimeters C = 28.27 inches = ? cm C = 28.27 inches = 71.8058 centimeters -- Earth's Polar Circumference: 40,008 km ; 24,859.82 mi -- NOTE: Activity can be done indoors using clamp-on lamps. (Bring/wear sunglasses!) >> Preparation & Provisioning: (x) - Check out "Cosmos" video; cue to 5-min. segment re: Eratosthenes Cosmos Episode #1 - Shores of the Cosmic Ocean [FCPS #29531] 27:58 - 28:31 Prelude: Nice montage of human faces & music 28:31 - 35:13 (6min42sec) Eratosthenes' segment (x) - Borrow: protractors; clay (Art/Sorenson); at least seven (7) basketballs &/or playground balls (P.E./Sorber) (x) - Cut straws & shorter pieces of string [alternate procedure: use paper (instead of string) to measure distance and trace shadow angle] [7.5" straws; cut into 5 pieces 1.5" long. Use OPAQUE straws (cast better shadows)] (x) - Cut longer piece(s) of string for measuring actual circumference of ball(s); calculate % error. Should get better results with larger balls; best results using BIG ball like Earth! What if you're off by 1 cm when measuring the circumference of a ball ~72 cm in circumference? Now imagine you're off by 1 cm in measuring the circumference of the Earth (C = 40,008 km or 4,000,800,000). In which scenario is the 1-cm error more significant? It's fairly obvious! == Related websites: ***http://k12science.ati.stevens-tech.edu/noonday/noon.html [follow series hyperlinks to GOOD STUFF, incl. QT mov of Sagan "Cosmos" segment, lots of good graphics, applets, etc. ***http://k12science.ati.stevens-tech.edu/noonday/cosmos.html [features link to QT mov] ***http://k12science.stevens-tech.edu/noonday/teacherguide.html Teacher Guide Measuring the Circumference of the Earth The Noon Day Project Google Search: eratosthenes + circumference Earth http://www.youth.net/eratosthenes/welcome.html == >> Eratosthenes' assumptions: (T) Sun's rays are parallel (T) Earth is round (T) Alternate interior angles of parallel lines are equal (F) Syene and Alexandria are on the same meridian >> Precautions: * "Sticks" (straws) are perpendicular to surface of ball, that is, if a line were extended from the straw it would intersect the center of the ball. * Measure shadow angle first * Measure distance (between sticks) from ~center of sticks >> Follow-Up: * Find the % error of your experimental results. Explain why your percent error is likely to be much higher than Eratosthenes'. * What are some possible sources of experimental error? * C = pi x d; solve for diameter (of ball). * B-ball as a scale model of the Earth (globe); find its scale. 1 ----- AS/MS Where: AS = Actual Size (circumference of Earth); MS = equivalent Model Size (circumference of b-ball) 1 // 40,008 km (convert to cm)/ ? cm (circumference of b-ball) == See NYSRE Earth Science Topic III for related P.O.D./quiz/test items. Earth Science Test (Topic III) [item from Topic III quiz] You have just landed on an unknown planet and attempt to determine the size of the planet as Eratosthenes did. The shadow angle is 1 degree, and the distance between the two stakes [sticks] is 25 km. What is the circumference of the planet? less than 1 km about 7 km A. 9,000 km cannot be determined from the data given 5. If you determine the circumference of a planet to be 6,300 km and subsequent measurements show the true value (TV) to be 10,000 km, what is your percentage error? 3,700% 63% A. 37% [% error = TV - EV / TV x 100] 3.7% Directions (9-10): Suppose you are measuring the circumference of the Earth and are not particularly careful about the measurements you make. For each statement in questions nine (9) through 10, write the letter of the word, chosen from the list below, that indicates how the calculated circumference of the Earth would be affected by the situation described. increased decreased unaffected 9. The distance between the two sticks is actually greater than your measurement. 10. Your reading of the angle of the shadow cast by the second stake is too large. [Topic III quiz version of #9 above] In your measurement of the circumference of the Earth, the distance between the two sticks is actually greater than your measurement. The calculated circumference would be decreased increased unaffected [Refer to graphic of globe from questions 11-16. Note: Locations B & D are on the same meridian; D is the more southerly of the two locations.] 16. The distance between locations D and B is approximately 3,000 km. When the Sun is directly overhead at location D, what other measurement would be necessary to calculate the Earth's circumference? the distance from location C to the Equator A. the angle of a shadow cast by a vertical pole at B the distance from D to E the time of day at location A 18. The circumference of the Earth is about 4.0 x 10(4) km. This value is equal to 400 km 4,000 km A. 40,000 km 400,000 km [Test item featuring graphic of Eratosthenes' method, 1982 JUN NYSRE, #7 (not available online).] 21. Based on the diagram at the right, what is the circumference of Planet X? [Note: Shadow angle = 20 degrees; distance = 1,000 km] 9,000 km A. 18,000 km [C = 360 x d / shadow angle] 24,000 km 36,000 km 26. Eratosthenes had to make several assumptions before he could use his method of measuring the circumference of the Earth. Which [one] of Eratosthenes' assumptions proved to be false? the Sun's rays are parallel alternate interior angles of parallel lines are equal the Earth is round Syene and Alexandria are on the same meridian 30. If the accepted average circumference of the Moon is 10,927 km, what is its approximate diameter? A. 3,480 km [D = C / pi] 10,002 km 6,595 km 2,734 km -- BRIES, CEBCO Standard Publishing p. 14 (graphic) shadow angle = 18deg; d = 2,000 km diff. in Sun's altitude (between two locations) = 90 - 72 = 18 degrees p. 24 (graphic of Planet Y) shadow angle = 25deg; d = 1,200 km ================================================================= Walter Sanford, Director Carl Sandburg Middle School Center for Sky Awareness 8428 Fort Hunt Road Sandburg Planetarium Alexandria, VA 22308 Fairfax County Public Schools Work: 703-799-6169 -6197 (fax) E-mail: wsanford@wsanford.com Home: 703-765-9392 AMS Project ATMOSPHERE Atmospheric Education Resource Agent & Water in the Earth System (WES) Resource Teacher SCSA, Geosystems, & Camp T-Equity - URL: http://www.wsanford.com/ =================================================================