GEOMETRY MAP4C

GEOMETRY The environment we live in is 3-dimensional and geometry is the natural language to express concepts and relationships of space. We give names to help us identify shapes in space (3D) and in a plane (2D). We classify objects according to certain attributes. We discover properties and relationships while dealing with various shapes and relate them to the world around us. We abstract plane figures from solid objects by seeing them as faces of solids that can be traced on paper.

GEOMETRY We name, classify, investigate and discover new properties for plane figures. We learn to construct plane figures and then the nets to construct solids. In this way, our knowledge of geometry grows in complexity following a 3- step progression: 3D (Manipulative, solids, concrete)  2D (Pictorial, graphic)  1D (Numerical, Abstract) It is this interplay of the real world of objects and the abstract world of thought that leads to progress.

SOLID OBJECTS: 3D, REAL Basic figures: sphere, cone, cylinder, prism, pyramid. Congruent figures: Spheres, pyramids …. Symmetries: Objects in space Stacking: Objects in space, packing Measurement: volume, surface area

PLANE FIGURES: 2D, FLAT Basic figures: point, line, ray, segment, angles Polygons: triangles, quadrilaterals….. Circles: centre, radius, diameter, circumference Congruent figures: triangles, squares…. Similar figures: map, scale drawing Measurement: perimeter, area

PRISMS A polyhedron with 2 congruent (equal in size and arrangement) faces (bases) that are parallel to each other. The remaining faces are parallelograms. A segment between bases is the altitude. Triangular PrismRectangular PrismPentagonal Prism

PYRAMIDS A polyhedron where all faces (but one) have a common vertex. The base is a polygon. Lateral faces are triangles. A line segment from the vertex to the base (perpendicular line) is the altitude. Square-based pyramid with vertical height (altitude).

SURFACE AREA To find the surface area of an object, it is useful to unfold the object on paper (or for real with a cereal box) and layout the 3D object as a 2D net. A net is useful to find the total surface area as the sum of the areas of all the faces.

EXAMPLE 1: RECTANGULAR PRISM SA = area of 2 bases and area of 2 sides and area of 2 ends. Expand the prism into a net (folded out flat) as shown: SA = 2((4x10) + 2(5x4) + 2(5x10) = 2(40) + 2(20) + 2(50) = = 220 cm 2

EXAMPLE 2: TRIANGULAR PRISM Shapes: 2 triangles, 3 different rectangles Need the other length of the triangle using Pythagoras = 13 cm. SA = 2 ( ½ )(5)(12) + 5(15) + 13(15) + 12(15) = = 510 cm 2

EXAMPLE 3: SQUARE BASED PYRAMID A square based pyramid has Surface Area = Area of base + Area of Lateral Faces If L=w = 5.0 mm, s = 10.0 mm, h = 6.0 mm, find SA. SA = Lw + 4(bs÷2) = (5)(5) + 4[(5)(10) ÷2] = (25) = 125 mm 2

SPHERE For a sphere, the SA is found as 4πr 2. A sphere has a radius of 25 cm. Find the surface area to the nearest hundredth. SA = 4(3.14) (25) 2 = 4(3.14) (625) = cm 2

CONE The surface area of a cone is the area of the base and the area of the unrolled peak. (lateral surface) SA cone = πr 2 + πrs Note: Pythagorean theorem may be used to find h, s or r. r s h

CLASSWORK Page 202 #2 (SA), 4, 7, 9, 14 Page 208 #2-6, 11, 13, 17