Listed are about 23 geometry problems which are actual S.A.T. problems or close facsimiles. I would welcome any interesting or challenging problems from geometry teachers,students or anyone else interested.
I introduce each class session with a "pre-class" problem. The instant the student gets to the classroom she or he knows they must be tried. After a time the students actually look forward to these problems. They are all closely related to S.A.T. problems and they are appropriate to the material covered in the previous classroom sessions.
These problems are very useful. The class always gets started quickly and time on task is maximized. Students are excited by the problems. The fact that they are from S.A.T. tests challenge and motivate the students. The concepts of geometry are reinforced. Since the problems involve several concepts, review is built into the process.
Geometry problems constitute 30% of the S.A.T. My feeling is that using this format helps the students to higher scores in the S.A.T. as well as a deeper understanding of the concepts of geometry.
1. If the perimeter of a square is 40 find its area. 2. If the area of a square is 144 find its perimeter. 3. If the circumference of a circle is 14p find its area. 4. If the area of a circle is 81p find its circumference. 5. If the ratio of the angles of a triangle is 2:3:5 find the measure of each angle. 6. If the ratio of 2 complementary angles is 2:3 find the measure of the larger angle. 7. If the ratio of 2 supplementary angles is 1:3 find the measure of the angles. 8. Find the area of an isosceles right triangle if its hypotenuse is 12. 9. Find the hypotenuse of an isosceles right triangle if the area of the triangle is 50. 10. If the perimeter of a square is 32 find the length of its diagonal. 11. If the diagonal of a square is 12, find the perimeter and the area of the square. 12. If the perimeter of an equilateral triangle is 36, find its area. 13. If the area of a square is 20 find the length of its diagonal. 14. Find the area of a sector of a circle if its central angle is 45 degrees and its diameter is 24. 15. The legs of a right triangle are in the ratio of 5:12. If the hypotenuse is 26, find the area of the triangle. 16. Find the ratio of a side of a square to its perimeter. 17. Find the ratio of a side of a square to its diagonal. 18. The ratio of the sides of 2 squares is 2:3. Find the ratio of their perimeters and the ratio of their areas. 19. A triangle has sides of 6,6,x. What are all numbers x for which the triangle is obtuse? 20. How many cubes with an edge of 2 will fit into a cube with an edge of 4? 21. A sector of a circle has a central angle of 120 degrees. if the area of the sector is 12p, find the radius of the circle. 22. A triangle has sides of 5,7,x. If x is an integer find its minimum and its maximum. 23. Find the area of a circle if its circumference is 16.