Regular Polygons

REGULAR POLYGONS A regular polygon is a polygon with all its sides equal and all its interior angles equal. All vertices of a regular lie on a circle whose center is the center of the polygon. Each side of a regular polygon subtends an angle θ = Also X = Y = 360 n = 180(n − 2) 2 2n 180 − 360 n at the centre, as shown in the figure. . Therefore, interior angle of a regular polygon = X + Y = Sum of all the angles of a regular polygon = n × 180(n − 2) n . 180(n − 2) = 180(n − 2) . n What is the interior angle of a regular octagon? Answer: The interior angle of a regular octagon = 180(8 − 2) = 135° 8 The formula for sum of all the angles of a regular polygon, i.e. 180(n − 2) , is true for all n-sided convex simple polygons. Let’s look at some polygons, especially quadrilaterals: Quadrilateral: A quadrilateral is any closed shape that has four sides. The sum of the measures of the angles is 360o. Some of the known quadrilaterals are square, rectangle, trapezium, parallelogram and rhombus. Square: A square is regular quadrilateral that has four right angles and parallel sides. The sides of a square meet at right angles. The diagonals also bisect each other perpendicularly. If the side of the square is a, then its perimeter = 4a, area = a2 and the length of the diagonal = 2a Rectangle: A rectangle is a parallelogram with all its angles equal to right angles. Properties of a rectangle: Sides of rectangle are its heights simultaneously. Diagonals of a rectangle are equal: AC = BD. A square of a diagonal length is equal to a sum of squares of its sides’ lengths, i.e. AC² = AD² + DC². Area of a rectangle = length × breadth Parallelogram: A parallelogram is a quadrangle in which opposite sides are equal and parallel. Any two opposite sides of a parallelogram are called bases, a distance between them is called a height. Area of a parallelogram = base × height Perimeter = 2(sum of two consecutive sides) Properties of a parallelogram: 1. Opposite sides of a parallelogram are equa l(AB = CD, AD = BC ). 2. Opposite angles of a parallelogram are equal ( A = C, B = D ). 3. Diagonals of a parallelogram are divided in their intersection point into two ( AO = OC, BO = OD ). 4. A sum of squares of diagonals is equal to a sum of squares of four sides: AC² + BD² = AB² + BC² + CD² + AD². Rhombus: If all sides of parallelogram are equal, then this parallelogram is called a rhombus. Diagonals of a rhombus are mutually perpendicular ( AC BD ) and divide its angles into two ( BCA, ABD = CBD etc. ). 1 1 Area of a rhombus = × product of diagonals = × AC × BD 2 2 DCA = Trapezoid: Trapezoid is a quadrangle two opposite sides of which are parallel. Here AD || BC. Parallel sides are called bases of a trapezoid, the two others (AB and CD ) are called lateral sides. A distance between bases (BM) is a height. The segment EF, joining midpoints E and F of the lateral sides, is called a midline of a trapezoid. A midline of a trapezoid is equal to a half-sum of bases: and parallel to them: EF || AD and EF || BC. A trapezoid with equal lateral sides ( AB = CD ) is called an isosceles trapezoid. In an isosceles trapezoid angles by each base, are equal ( A = D, B = C).. Sum of parallel sides AD + BC Area of a trapezoid = × height = × BM 2 2 In a trapezium ABCD with bases AB and CD , the sum of the squares of the lengths of the diagonals is equal to the sum of the squares of the lengths of the non-parallel sides and twice the 2 2 2 2 product of the lengths of the parallel sides: AC + BD = AD + BC + 2 ⋅ AB ⋅ CD Here is one more polygon, a regular hexagon: Regular Hexagon: A regular hexagon is a closed figure with six equal sides. If we join each vertex to the centre of the hexagon, we get 6 equilateral triangles. Therefore, if the side of the hexagon is a, each equilateral triangle has a side a. Hence, area of the regular hexagon = 3 2 3 3 2 6× a = a . 4 2