Measurement

Table of contents Revise the volume and surface areas for right prisms and cylinders Study the effect on volume and surface area when multiplying any dimension by a factor, k. Calculate the volume and surface areas of spheres, right pyramids and right cones.

Theory A polyhedron is a three dimensional figure with flat surfaces. The flat surfaces are called faces. Two faces intersect at an edge. Three or more edges intersect at a vertex. A regular Polyhedron is called a Platonic solid – all lateral surfaces are the same regular polygon and equal numbers of sides meet at each vertex. A prism is a polyhedron with two identical parallel bases. The bases must be polygons. The other faces are parallelograms and are called lateral faces. A prism is named by the shape of it’s base. In right prisms the lateral faces are perpendicular to the bases. In above cases all lateral faces are rectangles. A pyramid is a polyhedron with a polygon as base. The lateral sides on the polyhedron meet at a single vertex called the apex. A right prism is such that the apex is perpendicularly above the centre of the base.

Volume and Surface area of prisms and cylinders NameVolumeSurface Area Triangular Based Prism V = Area of base(triangle) x h Sum of 2 triangles and 3 rectangles Rectangular Prism / Cuboid V = Area of base(rectangle) x h Sum of areas of 6 rectangles Cube V = Area of base(square) x h Sum of areas of 6 squares Hexagonal Prism V = Area of base (hexagon) x h Sum of areas of 2 hexagons and 6 rectangles Cylinder V = Area of base(circle) x h 2πr²+2πrh

Volume and Surface area of Right Pyramids NameVolumeSurface Area Trianglular Based Pyramid ⅓ (A of Base) x h Sum of areas of base and 3 triangles Rectangular Based Pyramid ⅓ (A of Base) x h Sum of areas of base and 4 triangles Hexagonal Based Pyramid ⅓ (A of Base) x h Sum of areas of base and 6 triangles

Volume and Surface area of Cones and Spheres NameVolumeSurface Area Cone πr ³4πr² Sphere ⅓πr²h⅓πr²hπrl + πr²

Investigate the effect on the volume of a right prism If the dimensions of any right prism is increased by a factor k,then the volume will increase by a factor k 3 if k>1. If the dimensions of any right prism is increased by a factor k the volume will decrease by a factor k 3 if 0 < k < 1.

Formulae Volume of a cylinder V = л.r 2.h. These approximate values are often used for л: Л ≈ = 3 Л ≈ 3.14 or Л ≈ value on the calculator

Investigate the effect on the volume of a cylinder If the radius and height of a cylinder are each multiplied by a factor of k then the volume will increase by a factor k 3 if k>1. If the radius and height of a cylinder are each multiplied by a factor of k then the volume will decrease by a factor k 3 if 0<k<1.

Test your knowledge Question Determine the volume of a rectangle with a height = 4 cm, length = 8cm and breadth = 6 cm. Answer A) 264 cm 3 B) 192 cm 3 C) 184 cm 3 D) cm 3

Test your knowledge Question Determine the volume of the above rectangle if each side is multiplied by a scale factor of 3. Answer A) 1728 cm 2 B) 6315 cm 2 C) 5184 cm 2 D) 1469 cm 2

Investigate the effect on the surface area of a prism The surface area of a prism is the sum of the areas of it’s faces. Surface area of a rectangular prism = 2 ( l + b ). H + 2lb If all dimensions are multiplied by k, then the surface area will be multiplied by k 2. If the base or the perimeter is multiplied by k then the area multiplies by k.

Formulae A right cylinder has a surface area T, given by: T = 2лrh + 2лr 2 where h = height in units and r = radius in units These approximate values are often used for л: Л = =3 л = 3.14 or Л = value on the calculator

If the radius and the height of a cylinder are multiplied by a constant factor k, then the surface area of the cylinder becomes. Tnew = 2л(kr)(kh) + 2л(kr) 2 = k 2 (2лrh + 2 лr2) = k 2 Investigate the effect on the surface area of cylinder

Investigate the effect on the surface area of a cylinder If the radius and height of a cylinder are each multiplied by a factor of k then the surface area will increase by a factor k 2 if k>1. If the radius and height of a cylinder are each multiplied by a factor of k then the surface area will decrease by a factor k 2 if 0<k<1.

Test your knowledge Question Determine the surface area of a cylinder with a radius = 65 mm and height = 700mm. Answer A) cm 2 B) cm 2 C) cm 2 D) cm 2

Question Determine the surface area of the above cylinder if the radius and height is increased by a scale factor of 2. Answer A) cm 2 B) cm 2 C) cm 2 D) cm 2 Test your knowledge