Prepared by: RTFVerterra Plane and Solid Geometry Formulas ASIAN DEVELOPMENT FOUNDATION COLLEGE Given four sides a, b, c, d, and sum of two opposite angles: The content of this material is one of the intellectual properties of Engr. Romel Tarcelo F. Verterra of Asian Development Foundation College. Reproduction of this copyrighted material without consent of the author is punishable by law. (s a)(s b)(s c)(s d) abcdcos2 A= Tacloban City s= a b c d 2 = A C B D or = 2 2 A = ½ ab sin B + ½ cd sin D Polygons whose sides are equal are called equilateral polygons. Polygons with equal interior angles are called equiangular polygons. Polygons that are both equilateral and equiangular are called regular polygons. The area of a regular polygon can be found by considering one segment, which has the form of an isosceles triangle. Circumscribing x circle A = ab sin A C A B a Given three angles A, B, and C and one side a: a 2 sinB sinC A= 2 sin A The area under this condition can also be solved by finding one side using sine law and h b c d1 d2 D Area = d A A + C = 180° B + D = 180° Area = ½ C r POLYGONS d Area, A = a2 Perimeter, P = 4a There are two basic types of polygons, a convex and a concave polygon. A convex polygon is one in which no side, when extended, will pass inside the polygon, otherwise it called concave polygon. The following figure is a convex polygon. 4 a a Diagonal, d = a 2 General quadrilateral C b d1 2 d2 d Given diagonals d1 and d2 and included angle : A = ½ d1 d2 sin 5 3 c D A 4 3 B 2 1 1 5 = = = = = = = Circle circumscribed about a quadrilateral A circle is circumscribed about a quadrilateral if it passes through the vertices of the quadrilateral. b d (ab cd)(ac bd)(ad bc) r 180 C 2 r 360 r r triangle quadrangle or quadrilateral pentagon hexagon heptagon or septagon octagon nonagon C r 2 3 r a Aquad s c d ; s = ½(a + b + c + d) abcd SOLID GEOMETRY = 360 - r r POLYHEDRONS A polyhedron is a closed solid whose faces are polygons. bh h Ellipse b Area = a b Perimeter, P P = 2 r= Aquad = O Area = Asector + Atriangle Area = ½ r2 r + ½ r2 sin Area = ½ r2 (r + sin ) Area = A circle is inscribed in a quadrilateral if it is tangent to the three sides of the quadrilateral. O r = angle in radians 6 (s a)(s b)(s c)(s d) Circle incribed in a quadrilateral b Area = Asector – Atriangle Area = ½ r2 r – ½ r2 sin Area = ½ r2 (r – sin ) r 6 r c a D Segment of a circle Parabolic segment Polygons are classified according to the number of sides. The following are some names of polygons. 3 sides 4 sides 5 sides 6 sides 7 sides 8 sides 9 sides b ra = A T ; rc = A T ; rb = A T s a s c s b s = ½(a + b + c + d) r Note: 1 radian is the angle such that C = r. Square c a 4Aquad Sector of a circle Area = ½ r2 radians = C b Circles escribed about a triangle (Excircles) A circle is escribed about a triangle if it is tangent to one side and to the prolongation of the other two sides. A triangle has three escribed circles. Aquad = Arc C = r radians = r r A r= Area, A = r2 = D2 4 a b c d 2 “For any cyclic quadrilateral, the product of the diagonals equals the sum of the products of the opposite sides” d1 d2 = ac + bd a2 b2 Perimeter, P = n x n 2 Interior angle = n 180° Circumference = 2r = D (s a)(s b)(s c)(s d) a r ra Circle Ptolemy’s theorem Perimeter, P = 2(a + b) c Exterior angle = 360° / n a a circle. AT s s = ½(a + b + c) ra = 360° / n Area, A = ½ R2 sin n = ½ x r n C b Circle inscribed in a triangle (Incircle) A circle is inscribed in a triangle if it is tangent to the three sides of the triangle. B Incenter of the triangle x x = side = angle subtended by the side from the center R = radius of circumscribing circle r = radius of inscribed circle, also called the apothem n = number of sides a Cyclic Quadrilateral A cyclic quadrilateral is a B quadrilateral whose vertices lie on the circumference of s= b c ra x a b A= h 2 Rectangle a abc 4A T circle Apothem x A = a2 sin A Trapezoid apply the formula for two sides and included angle. d r= r a b r Given side a and one angle A: a b c 2 The area under this condition can also be solved by finding one angle using cosine law and apply the formula for two sides and included angle. Circumcenter of the triangle Inscribed R R A = ½ d1 d2 Given two sides a and b and included angle : A = ½ ab sin a x x A a Given diagonals d1 and d2: A = ½ bh Diagonal, d = d2 90° Given base b and altitude h Area, A = ab D d1 Given three sides a, b, and c: (Hero’s Formula) A = s(s a)(s b)(s c) A circle is circumscribed about a triangle if it passes through the vertices of the triangle. r= Regular polygons Given two sides a and b and one angle A: Rhombus s= b D a Given diagonals d1 and d2 and included angle : A = ½ d1 d2 sin b d2 A B C C d1 c Circle circumscribed about a triangle (Cicumcircle) Number of diagonals, D The diagonal of a polygon is the line segment joining two non-adjacent sides. The number of diagonals is given by: n D = (n 3) 2 Parallelogram h RADIUS OF CIRCLES AT = area of the triangle PLANE GEOMETRY a decagon undecagon dodecagon quindecagon hexadecagon Sum of exterior angles The sum of exterior angles is equal to 360°. = 360° Divide the area into two triangles B PLANE AREAS = = = = = Sum of interior angles The sum of interior angles of a polygon of n sides is: Sum, = (n – 2) 180° Given four sides a, b, c, d, and two opposite angles B and D: Part of: Plane and Solid Geometry by RTFVerterra © October 2003 Triangle 10 sides 11 sides 12 sides 15 sides 16 sides PRISM a2 b2 2 a b b a A prism is a polyhedron whose bases are equal polygons in parallel planes and whose sides are parallelograms. Prisms are classified according to their bases. Thus, a hexagonal prism is one whose base is a Prepared by: RTFVerterra Plane and Solid Geometry Formulas hexagon, and a regular hexagonal prism has a base of a regular hexagon. The axis of a prism is the line joining the centroids of the bases. A right prism is one whose axis is perpendicular to the base. The height “h” of a prism is the distance between the bases. Like prisms, cylinders are classified according to their bases. Fixed straight line ELLIPSOID Z Azone = 2rh 2 h Volume = (3r h) 3 Directrix b a a Spherical segment of two bases h r As = 2rh Ab Volume = Ab h Volume = Volume = Ab h d1 b a Cube (Regular hexahedron) Volume = Ab h = a3 Lateral area, AL = 4a2 Total surface area AS = 6a2 Face diagonal a d1 = a 2 Space diagonal d2 = a 3 d2 d1 a r Ab 1 r Vwedge n Similar to prisms, pyramids are classified according to their bases. Vertex Ab = area of the base h = altit ude, perpendicular distance fr om o the vertex t the base A2 L C R r The surface area generated by a surface of revolution equals the product of the length of the generating arc and the distance traveled by its centroid. Spherical segment of one base h r As = L 2 R Second proposition of Pappus h r 1 m r The volume area generated by a solid of revolution equals the product of the generating area and the distance traveled by its centroid. Volume = A 2 R 2 SIMILAR SOLIDS Two solids are similar if any two corresponding cubes are similar. First proposition of Pappus 3 3 A = 4r2 Surface area, s r 4A A x2 x2 x1 x2 x1 A cylinder is the surface generated by a straight line intersecting and moving along a closed plane curve, the directrix, while remaining parallel to a fixed straight line that is not on or parallel to the plane of the directrix. A Axis of rotation cg SPHERE A 1A 2 L The prismoidal rule gives precise values of volume for regular solid such as pyramids, cones, frustums of pyramids or cones, spheres, and prismoids. r x1 r L/2 6 x1 A2 L/2 Volume = E = spherical excess of the polygon E = sum of angles – (n – 2)180° r 3E Volume = 540 L = slant height = h2 (R r)2 h 2 2 Volume = R r Rr 3 Lateral area = (R + r) L Volume = 3 Am B SOLID OF REVOLUTION 4 3/2 L r R A1 CYLINDERS c D r PRISMOIDAL RULE C A 3 h h 3 b Spherical pyramid r h A A 1 2 r R = lower base radius A2 r2h 2 r 4r r 2 2 AL = h 2 3h2 4 r E 180 E = sum of angles – (n – 2)180° Frustum of right circular cone A frustum of a pyramid is the volume included between the base and a cutting plane parallel to the base. A1 = lower base area A2 = upper base area h = altitude 1 Volume = Area = h A A A 1 A 2 2 1 h A h h A1 Frustum of pyramid Volume = 270 B D A1 h 1e d Volume = ln e r 3 2 Frustum of a cone A1 = lower base area A2 = upper base area h = altitude Ab 3 360 A r 2 h2 L = slant height = 1 2 1 r h Ab h = Volume = 3 3 Lateral area, AL = r L A pyramid is a polyhedron with a polygonal base and triangular faces that meet at a common point called the vertex. 1 Vwedge = a r 2 PARABOLOID OF REVOLUTION 90 4 3 3 r h3 h1 h r Alune = A spherical polygon is a polygon on the surface of a sphere whose sides are arcs of great circles. n = number of sides; r = radius of sphere E = spherical excess L h 2 a b 2 4r 360 = 3 As = 2a2 + 2 = 4 Volume = Spherical polygons 3 Wedge Ab h a2 b2 / a e= Oblate spheroid A lune h4 Volume = A zone r = 2 r 2h 3 Prolate spheroid is formed by revolving the ellipse about its minor (Z) axis. Thus from the figure above, c = a, then, Directrix a Right circular cone r = base radius h = altitude PYRAMIDS 3 Spherical lune and wedge h AR = area of the right section n = number of sides Volume = AR Prolate spheroid is formed by revolving the ellipse about its major (X) axis. Thus from the figure above, c = b, then, 4 2 Volume = ab 3 arcsine As = 2b2 + 2ab e r Lune Generator Volume = h2 1 Volume = Similar to pyramids, cones are classified according to their bases. Vertex Ab = base area h = altitude Truncated prism AR b point, the vertex, and moving along a fixed curve, the directrix. a2 b2 c 2 Space diagonal, d2 = 4 abc 3 Prolate spheroid r A cone is the surface generated by a straight a 2 c 2 Face diagonal, d1 = (3a 3b h ) r CONE Volume = Ab h = abc Lateral area, AL = 2(ac + bc) 2 h h Lateral area, AL AL = Base perimeter h AL = 2 r h c 2 r Volume = Ab h = r2 h d2 6 Volume = 2 Spherical cone or spherical sector Right circular cylinder Rectangular parallelepiped h Y h Ab h X c x2 For all similar solids: 2 x = 1 x As2 2 As1 3 and 1 V V2 1 = x x 2 Where As is the surface, total area, or any corresponding area. The dimension x may be the height, base diameter, diagonal, or any corresponding dimension.