Chapter 1: Symmetry and Surface Area

Transcription

1 Chapter 1: Symmetry and Surface Area Name: Section 1.1: Line Symmetry Line of symmetry(or reflection): divides a shape or design into two parts. Can be found using: A mirra Folding Counting on a grid

2 Section 1.2 Rotational Symmetry Rotational Symmetry: when a shape or design can be turned about is so that it fits onto its outline more than once in a complete. Order of Rotation: Number of times a shape in one complete turn. Angle of Rotation = 1 or Order of rotation Angle of Rotation = 360 o Order of rotation Shape Order of Rotation 2 Angle of Rotation (Fraction) 1/2 Angle of Rotation (Degrees) = 180 o Lines of Symmetry 0 SHAPES CAN HAVE EITHER LINEAR SYMMETRY, ROTATIONAL SYMMETRY, OR BOTH!!!

3 Section 1.3 Surface Area Surface Area: of the areas of all the faces of an object (or, the amount of material needed to an object) Determine the and of each face Calculate the of each face using one of these formulas: Shape Formula in Words Formula in Symbols Square Area of a square = side length x side length A = s 2 Rectang le Area of a rectangle = length x width A = l x w Triangle Area of a triangle = (base x height) 2 A = b x h 2 Circle Area of a circle = pi x radius squared * pi = 3.14 A = r 2

4 Calculating the Surface Area of a Rectangular Prism Top and Bottom = l x w Front and Back Side and Side = l x h = w x h SA = Top and Bottom + Front and Back + Side and Side SA = 2 x ( l x w) + 2 x (l x h) + 2 x (w x h) = 2 x [(l x w) + (l x h) + (w x h)] TRY:

5 Calculating the Area of a Triangular Prism A triangular prism is made up of two bases and three. SA = Area of 2 triangles + A R #1 + A R #2 + A R #3 SA = 2 x (b x h) 2 + l x b + s 1 x l + s 2 x l TRY:

6 Calculating the Surface Area of a Cylinder A cylinder is made up of two bases and a strip with one dimension that is the of the cylinder and the other that is equal to the of the base. SA = 2 circular bases + (circ. of base) x height of cylinder SA = 2 x ( r 2 ) + 2 r x h TRY:

7 CONES: To calculate the surface area of a cone, you need: - the radius of the base - the slant height of the cone From the net, you can construct the large circle using the vertex of the cone as the centre and the slant height of the cone as the radius. The lateral area ( side ) of the cone forms a sector of the large circle. The circumference of the base of the cone, 2πr, forms the length of the arc AB of the sector. The circumference of the large circle that is formed from the lateral area is 2πs.

8 To determine the lateral area of the cone, you can set up a proportion of corresponding ratios: Lateral area of cone = Area of Large Circle Circumference of Cone Circumference of Large Circle Lateral Area of cone = πs 2 Lateral Area of cone = πs 2 2πr 2πs r s Lateral Area of cone = (r)( πs 2 ) s Lateral Area of cone = πrs This area, along with the area of the base of the cone, creates the formula for the surface area of a cone: SA cone = πr 2 + πrs Example #1: Calculate the surface area of the following cone:

9 Example #2: Calculate the surface are of the following cone: SPHERES: The surface area of a sphere, when flattened, can be modeled as a lateral strip wrapped around the sphere. Think of wrapping a right cylinder around a sphere. The diameter of the sphere would be the height of the cylinder, which is 2π. The circumference of the sphere would be the circumference of the cylinder, which is 2πr. The area of the lateral strip = length x width = (2πr)(2r) = 4πr 2 Therefore, the area of a sphere is: SA sphere = 4πr 2 Examples: Calculate the surface area of the following spheres:

10 PYRAMIDS: The surface area of a pyramid can be found by adding up the areas of each of the individual faces. For a rectangular prism: There are five faces: the rectangular base and four triangles. A BASE = w x l A Triangle #1 = (w x s 1 )/2 A Triangle #3 = (w x s 1 )/2 A Triangle #2 = (l x s 2 )/2 A Triangle #4 = (l x s 2 )/2 THEREFORE, SA rect.pyramid = w x l + (2)(( wx s 1 )/2) + (2)((l x s 2 )/2) SA rect.pyramid = w x l + w x s 1 + l x s 2 Example 1: Sketch a square pyramid with a base measuring 10 cm on each side. The slant height of each face is 8.5 cm. What is the surface area of the pyramid?

11 Example #2: Calculate the surface area of the following rectangular prism (hint: Find the slant heights first!) Example #3: Calculate the surface area of the following triangular prism.

12 Surface Area of Compound Shapes Consider how the shape is made from its Determine the of each part. Subtract the area of any surface. TRY: Note: Surface Area of a Sphere = 4 r 2 TRY: