Chapter 1 - Symmetry and surface area
Symmetry and surface area objectives:
-Finding symmetry
-Using lines of symmetry
-Rotation symmetry
-Rotating shapes
-Create designs with symmetry
-Using symmetry to find surface area
1.1 Lines of symmetry
A line of symmetry is a line that divides a figure into two reflected parts. It is sometimes called a line of reflection or axis of symmetry.
NOTE: A figure may have one or more lines of symmetry, or it may have none. Lines of symmetry can be vertical, horizontal or oblique (slanted). Identical halves can be reflected in a vertical, horizontal or oblique line.
-Finding symmetry
-Using lines of symmetry
-Rotation symmetry
-Rotating shapes
-Create designs with symmetry
-Using symmetry to find surface area
1.1 Lines of symmetry
A line of symmetry is a line that divides a figure into two reflected parts. It is sometimes called a line of reflection or axis of symmetry.
NOTE: A figure may have one or more lines of symmetry, or it may have none. Lines of symmetry can be vertical, horizontal or oblique (slanted). Identical halves can be reflected in a vertical, horizontal or oblique line.
The lines in the Triangle are examples of vertical and oblique lines of symmetry.
Line 1 is Vertical
Line 2 is Oblique
Line 3 is also Oblique (slanted)
Line 1 is Vertical
Line 2 is Oblique
Line 3 is also Oblique (slanted)
The lines in this square are Oblique, Horizontal and vertical
Line 1 is Horizontal
Line 2 is Oblique
Line 3 is Vertical
Line 4 is Oblique
Line 1 is Horizontal
Line 2 is Oblique
Line 3 is Vertical
Line 4 is Oblique
The shape on the left has no lines of symmetry while the three on the right have one, three and infinite.
You can complete a symmetric drawing by folding or reflecting one half in the line of symmetry. The opposite halves are mirror images.
1.2 Rotation symmetry and transformations
Transformations
-A transformation moves a geometric figure
-Translations, reflections, rotations
-A transformation moves a geometric figure
-Translations, reflections, rotations
Center of Rotation
-The center of rotation in rotation symmetry is the point about which the rotation of an object or design turns
Rotation Symmetry
-This is when a shape or design can be turned about its center of rotation so that it fits onto its outline more than once in its turn.
Order of Rotation
- the number of times a shape or design fits into itself in one complete 360 degree turn
Angle of Rotation
-The minimum measure of the angle needed to turn a shape or design onto itself
May be measured in degrees or fractions of a turn
= 360 degrees / order of rotation
For example, the center of rotation is O. The order of rotation is 1, because if it completes a 360 degree turn, the shape fits into itself only once. So the angle of rotation is 360 degrees / 1
-The center of rotation in rotation symmetry is the point about which the rotation of an object or design turns
Rotation Symmetry
-This is when a shape or design can be turned about its center of rotation so that it fits onto its outline more than once in its turn.
Order of Rotation
- the number of times a shape or design fits into itself in one complete 360 degree turn
Angle of Rotation
-The minimum measure of the angle needed to turn a shape or design onto itself
May be measured in degrees or fractions of a turn
= 360 degrees / order of rotation
For example, the center of rotation is O. The order of rotation is 1, because if it completes a 360 degree turn, the shape fits into itself only once. So the angle of rotation is 360 degrees / 1
For example, p is the center of rotation, If the angle of rotation is 360 degrees, then the order of rotation would be 4, because it fits onto its outline more than once in its rotation.
Rotation SYmmetry
Order of rotation Angle or rotation (Degrees) Angle or rotation (Fraction of turn)
a) 2 360/2 = 180 degrees 1 turn/2 = 1/2 turn
b) 5 360/5 = 72 degrees 1 turn/2 = 1/2 turn
c) 1 360 degrees 1 turn
a) 2 360/2 = 180 degrees 1 turn/2 = 1/2 turn
b) 5 360/5 = 72 degrees 1 turn/2 = 1/2 turn
c) 1 360 degrees 1 turn
1.3 Surface area
Surface Area - The sum of the areas of all the faces of an object
Circle Area = 3.14r2 since there is both a top and bottom, that gets multiplied by two.
Rectangular prism = 2 times the area of the base plus area of the 3 rectangular faces
Cube area = 6a2
Circle Area = 3.14r2 since there is both a top and bottom, that gets multiplied by two.
Rectangular prism = 2 times the area of the base plus area of the 3 rectangular faces
Cube area = 6a2
Cylinders
Areas of the top, bottom and side (s). Circle area formula is Pir2 (3.14 times radius squared) Since there is both a top and a bottom, that gets multiplied by two. Think of a soup can, the area of a rectangle is the circumference of the circle x height or (2 pi r 2) x H Surface Area = (2Pir 2) + Pi d h |
Cube
-Surface area of a cube is the area of the six squares that cover it -The area of one of them is multiplied by a (a x a), or a2 -Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared. Surface area = 6s2 |
Rectangular Prism
-The surface area of a rectangular prism is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same -The area of the top and bottom (side lengths a and c) = a multiplied by c (a x c) -Since there are two of them, you get 2ac -The front and back have side lengths of b and c. The area of one of them is multiplied by c (b x c) and there are two of them, so its 2bc -The left and right side have side lengths of a and b, so the surface area of one of them is a multiplied by b, (a x b). So its 2ab again. |
Area of top and bottom:
4 x 4=16
Area of front and back:
4 x 8=32
There are two different ways to figure out surface area of this prism
4 x 4=16
Area of front and back:
4 x 8=32
Area of left and right side: 8 x 4= 32
There are two different ways to figure out surface area of this prism
1) Multiply 2 by each sides area (l x w) and then add all 3 totals together
Ex. 2(4x4) + 2(4x8) + 2(8x4) = 160
OR
2) Find each sides area, add the three totals together then multiply the total by two
Ex. (4x4) + (4x8) + (8x4) = 80