Perimeter, Area and Volume of Regular Sapes Perimeter of Regular Polygons Perimeter means te total lengt of all sides, or distance around te edge of a polygon. For a polygon wit straigt sides tis is te sum of all sides. Eg. triangle rectangle parallelogram trapezium 11 8 9 5 5 6 6 7 7 4 4 4 11 8 5 + 5 + 4 = 14cm 6 + 6 + 11 + 11 = 4cm 8 + 8 + 7 + = 0cm 4 + 4 +9 + = 0cm All dimensions given in cm (not drawn to scale) For polygons wit curved sides te perimeter is known as te circumference and is given by te formula Were Eg. Circumference = πr for a circle and π a² + b² for an ellipse radius = 5 π is a matematical constant wit te value of.14 (correct to decimal places) r is te radius of te circle (distance from centre to circumference) a is te major radius of an ellipse. b is te minor radius of an ellipse.._.._.._.._.._ a = 6cm b = 4cm Circumference = πr x.14 x 5 1.4cm circumference = x.14 6 + 16 Area of Regular Polygons Te area of a polygon is te space it occupies in a single plane. For squares, rectangles and parallelograms te area is given by Area = base x eigt =.04cm Eg. 1 4 t = 7 1 1 8 1 x 1 = 144cm² 1 x 4 = 48cm² 8 x 7 = 56cm² Heigt is defined as te perpendicular distance between te pair of parallel sides
All dimensions given in cm (not drawn to scale) For Triangles area = ½ x base x eigt Were eigt is distance from apex to meet base at rigt angle cm Area = ½ x 1 x = 18cm² For Trapeziums 8 cm area = ½ sum of parallel sides x eigt 7 cm Area = ½ x (8 + 14) x 7 = 77cm² 14 cm For Circles area = π r² Area = π r² r = 5 =.14 x 5² = 78.54cm² For a sector of a Circle r = 5 60 area = area of circle x sector angle 60 Area of sector = π r² x 60 = 1.1cm² 60 For Ellipse b = 5 area = π ab Area =.14 x 10 x 5 = 157cm² a = 5 Complex sapes for wic tere are no formulas sould be divided into simple sapes. Te area of eac is ten calculated and added togeter to determine te overall area. Area = A1 + A + A + A4 - A5 A1 A A5 A A4
Volume of Regular Sapes Volume is te amount of space in dimensions occupied by a sape. Prism A prism is any sape were te cross-sectional area is constant. For any prism: Volume = area of base x eigt Rectangular Prism area of base = lengt x breadt volume = lengt x breadt x eigt I b Saded area is te base eg. calculate te volume of a block wit a square base of side 6cm and a eigt of 10cm volume = 1 x b x = 6 x 6 x 10 = 60cm³ Triangular Prism area of base = ½ x base x eigt 1 b Saded area is te base volume = ½ x base x 1 x eg. determine te volume of a component 16cm long wit a triangular cross-section wic as a base of 4cm and perpendicular eigt of 5cm area of base = ½ x 4 x 5 volume = ½ x 4 x 5 x 16 = 160cm³ Circular Prism area of base = π r² volume = π r² x eigt r Saded area is te base eg. calculate te volume of a cylinder wit a radius of 5cm and a eigt of 4cm. volume = π r² x eigt =.14 x 5² x 4 = 14.cm³
Te volume of certain non-prismatic sapes can be determined by using te correct formula. Spere r Pyramid and cone volume of a spere = 4 π r eg. determine te volume of a sperical component wit te radius of 7cm. volume = 4 x.14 x 7³ = 146.76cm³ 1 r b volume = 1 x base area x eigt Pyramid volume = 1 x 1 x b x Cone volume = 1 x π r² x eg. calculate te volume of a cone wit base radius of 6cm and perpendicular eigt of 10cm Volume = 1 x.14 x 6² x 10 = 76.00cm³ Volumes of irregular sapes can be determined by calculation if te mass and density of te material from wic it is known or by displacement. Calculation of volume using density and mass. eg. density of substance from wic an irregular object is made is 8500kg/m³. if it as a mass of 45kg, calculate its volume. Volume = mass = 45 = 0.05m³ density 8500 Measurement of volume using displacement 500cc nd reading 00cc 1st reading volume = nd reading 1st reading = 500 00 = 00cc Measuring cylinder
Perimeter, Area and Volume of Regular Sapes Workseet 1 Calculate te area of te following sapes 1... 5 cm 15 cm 9.5 cm 15 cm 4. 1. cm cm 5..5 cm 7.8 cm 4.5 cm 1. cm 47.5 cm 6 cm.5 cm 4 cm 5 cm 6. A water tank is a cuboid wit a base of 1.m by 0.8m. How deep is te water wen te tank contains 0.84m³ of water? 7. A classroom is 5m x 6m x m. Healt regulations require tat eac student must ave a minimum of 5m³ of air. How many students can occupy te room? Calculate te volume of te following sapes. All dimensions in cm. 8. 9. 10. Internal r = 0.75 External r = 1.00 1 8 15 6
Perimeter, Area and Volume of Regular Sapes Workseet Calculate te saded area of te following sapes 1... 5 cm cm 6 cm 9 cm.5 cm cm 4. 5. 15 cm all circles ave a radius of 1.5cm 4 cm cm.5 cm 6 cm 16 cm 9 cm cm cm radius of circle = cm 6. An ingot 80 x 10 x 00mm is cast into a cylinder 10mm diameter. Calculate its lengt. 7. A rivet as a emisperical ead 6mm radius and a stem of 6mm diameter and 15mm lengt. Calculate te volume of 100 of te rivets. 8. Wat would be te volume of (a) air (b) plastic in a ball wit 5cm diameter made from plastic mm tick? Calculate te volume of te following sapes. All dimensions in cm. 7 9. 10. 5 Normal t = 6 14 Cylinder radius = 1.5 5 5