■ A central angle of a circle is an angle that has its vertex at the center and that has sides that are radii. ■ Central angles have the same degree measure as the arc it forms. LENGTH OF ARC To find the length of an arc, multiply the circumference of the circle, 2πr,where r ϭ the radius of the circle, by the fraction ᎏ 36 x 0 ᎏ ,where x is the degree measure of the arc or central angle of the arc. Example: Find the length of the arc if x ϭ 36 and r ϭ 70. L = ᎏ 3 3 6 6 0 ᎏ × 2(π)70 L = ᎏ 1 1 0 ᎏ × 140π L = 14π A REA OF A SECTOR A sector of a circle is a region contained within the interior of a central angle and arc. A B C shaded region = sector r x r o – THE GRE QUANTITATIVE SECTION– 190 – THE GRE QUANTITATIVE SECTION– 191 The area of a sector is found in a similar way to finding the length of an arc. To find the area of a sector,sim- ply multiply the area of a circle, πr 2 , by the fraction ᎏ 36 x 0 ᎏ , again using x as the degree measure of the central angle. Example: Given x = 60 º and r = 8, find the area of the sector: A = ᎏ 3 6 6 0 0 ᎏ ϫ ( )8 2 A = ᎏ 1 6 ᎏ ϫ 64( ) A = ᎏ 6 6 4 ᎏ ( ) A = ᎏ 3 2 2 ᎏ ( ) Polygons and Parallelograms A polygon is a closed figure with three or more sides. TERMS RELATED TO POLYGONS ■ Ve rtices are corner points, also called endpoints, of a polygon. The vertices in the previous polygon are A, B, C, D, E, and F. ■ A diagonal of a polygon is a line segment between two nonadjacent vertices. The two diagonals indi- cated in the previous polygon are line segments BF and AE. ■ A regular (or equilateral) polygon’s sides are all equal. ■ An equiangular polygon’s angles are all equal. FE D C B A r x r o – THE GRE QUANTITATIVE SECTION– 192 ANGLES OF A QUADRILATERAL A quadrilateral is a four-sided polygon. Since a quadrilateral can be divided by a diagonal into two trian- gles, the sum of its angles will equal 180 + 180 = 360 degrees. INTERIOR ANGLES To find the sum of the interior angles of any polygon, use this formula: S = 180(x – 2) where x is the number of polygon sides. Example: Find the sum of the angles in the following polygon. S = (5 – 2) ϫ 180 S = 3 ϫ 180 S = 540 EXTERIOR ANGLES Similar to the exterior angles of a triangle, the sum of the exterior angles of any polygon equals 360 degrees. 5 1 2 3 4 1 2 3 4 m∠1 + m∠2 + m∠3 + m∠4 = 360° SIMILAR POLYGONS If two polygons are similar, their corresponding angles are equal and the ratio of the corresponding sides is in proportion. Example: These two polygons are similar because their angles are equal and the ratio of the corresponding sides is in proportion. Parallelograms A parallelogram is a quadrilateral with two pairs of parallel sides. In this figure, A ෆ B ෆ ʈ C ෆ D ෆ and B ෆ C ෆ ʈ A ෆ D ෆ . A parallelogram has the following characteristics: ■ Opposite sides are equal (AB = CD and BC = AD). ■ Opposite angles are equal (mЄA = mЄC and mЄB = mЄD). ■ Consecutive angles are supplementary (mЄA + mЄB = 180 º ,mЄB + mЄC = 180 º , mЄC + mЄD = 180 º ,mЄD + mЄA = 180 º ). ■ Diagonals bisect each other. SPECIAL TYPES OF PARALLELOGRAMS There are three types of special parallelograms: ■ A rectangle is a parallelogram that has four right angles. D A B C 60° 10 4 6 18 120° 60° 120° 5 2 3 9 – THE GRE QUANTITATIVE SECTION– 193 ■ A rhombus is a parallelogram that has four equal sides. ■ A square is a paralleloram in which all angles are equal to 90 degrees and all sides are equal to each other. DIAGONALS In all parallelograms, diagonals cut each other in two equal halves. ■ In a rectangle, diagonals are the same length. ■ In a rhombus, diagonals intersect to form 90-degree angles. DC AB AC = DB D CB A AB = BC = CD = DA ∠A = m∠B = m∠C = m∠D m D C B A AB = BC = CD = DA D A B C AB = CD BC = AD m∠A = m∠B = m∠C = m∠D – THE GRE QUANTITATIVE SECTION– 194 ■ In a square, diagonals have both the same length and intersect at 90-degree angles. Solid Figures, Perimeter, and Area You will need to know some basic formulas for finding area, perimeter, and volume on the GRE. It is impor- tant that you can recognize the figures by their names and understand when to use which formula. To begin, it is necessary to explain five kinds of measurement: P ERIMETER The perimeter of an object is simply the sum of the lengths of all its sides. 6 7 4 10 Perimeter = 6 + 7 + 4 + 10 = 27 B C A D AC = DB and AC DB B C A D BD AC – THE GRE QUANTITATIVE SECTION– 195 . QUANTITATIVE SECTION 19 0 – THE GRE QUANTITATIVE SECTION 19 1 The area of a sector is found in a similar way to finding the length of an arc. To find the area of a sector,sim- ply multiply the area. circle, 2πr,where r ϭ the radius of the circle, by the fraction ᎏ 36 x 0 ᎏ ,where x is the degree measure of the arc or central angle of the arc. Example: Find the length of the arc if x ϭ 36 and. equal 18 0 + 18 0 = 360 degrees. INTERIOR ANGLES To find the sum of the interior angles of any polygon, use this formula: S = 18 0(x – 2) where x is the number of polygon sides. Example: Find the sum