Two-Dimensional Figures, Transformations, and Three-Dimensional Solids

1-6: Two-Dimensional Figures and 10-2: Areas of Trapezoids

• Application: Ella's Designer Handbags

  • Ella has turned her hobby of making designer handbags and totes into a small business.
  • Among her designs is a trapezoid-shaped handbag.
  • To estimate the amount of material needed to produce each handbag, she must calculate the area of a trapezoid.

• Polygons and Classification

  • Classification Criteria: Polygons are classified by their number of sides, whether they are convex or concave, and whether they are regular or irregular.
  • Convex vs. Concave:
    • Convex: A polygon where no line segment between two points on the boundary ever goes outside the polygon.
    • Concave: A polygon that has at least one interior angle greater than $180^{\circ}$, effectively looking like a portion of the figure is "caved in."
  • Regular vs. Irregular:
    • Regular: A convex polygon that is both equilateral (all sides equal) and equiangular (all angles equal).
    • Irregular: A polygon that does not have all sides and all angles equal.

• Key Geometric Formulas (Perimeter, Circumference, and Area)

  • Triangle:
    • Perimeter: $P = a + b + c + d$ (Note: usually $a+b+c$ for triangles, transcript uses four variables for general polygon perimeter).
    • Area: $A = \frac{1}{2}bh$
    • Variables: $b$ is base, $h$ is height.
  • Square:
    • Perimeter: $P = s + s + s + s = 4s$
    • Area: $A = s \times s = s^{2}$
    • Variables: $s$ is side length.
  • Rectangle:
    • Perimeter: $P = l + w + l + w = 2l + 2w$
    • Area: $A = l \times w$
    • Variables: $l$ is length, $w$ is width.
  • Circle:
    • Circumference: $C = 2\pi r$ or $C = \pi d$
    • Area: $A = \pi r^{2}$
    • Variables: $r$ is radius, $d$ is diameter.
  • Trapezoid:
    • Area: $A = \frac{1}{2}h(b_{1} + b_{2})$
  • Calculation Note: The most accurate way to perform a calculation with pi is to use the $\pi$ symbol on a calculator. Use $3.14$ as an estimate ONLY if a calculator is unavailable.

• Practice Examples: Perimeter and Area

  • Mrs. Moore's Classroom Tape:
    • Goal: Use most or all of $19\,ft$ of tape to mark an area.
    • Option A: Square with side length $5\,ft$. $P = 4(5) = 20\,ft$ (Too much tape).
    • Option B: Circle with radius $3\,ft$. $C = 2\pi(3) \approx 18.85\,ft$ (Best fit).
    • Option C: Right triangle with legs of $6\,ft$. $P = 6 + 6 + \sqrt{6^{2} + 6^{2}} = 12 + 8.48 = 20.48\,ft$ (Too much tape).
    • Option D: Rectangle with length $8\,ft$ and width $3\,ft$. $P = 2(8) + 2(3) = 22\,ft$ (Too much tape).
  • Trapezoid Base Calculation:
    • Given: Area $A = 549\,cm^{2}$, height $h = 18\,cm$, and base $b_{2} = 35\,cm$.
    • Formula: $549 = \frac{1}{2}(18)(b_{1} + 35)$.
    • Simplify: $549 = 9(b_{1} + 35)$.
    • Divide by $9$: $61 = b_{1} + 35$.
    • Solve for $b_{1}$: $b_{1} = 26\,cm$.
  • Perimeter of Quadrilateral on Coordinate Plane:
    • Vertices: $R(-1, 3)$, $S(3, 3)$, $T(5, -1)$, and $U(-2, -1)$.
    • Length $RS = |3 - (-1)| = 4$.
    • Length $ST = \sqrt{(5-3)^{2} + (-1-3)^{2}} = \sqrt{2^{2} + (-4)^{2}} = \sqrt{4 + 16} = \sqrt{20}$.
    • Length $UT = |5 - (-2)| = 7$.
    • Length $RU = \sqrt{(-2 - (-1))^{2} + (-1 - 3)^{2}} = \sqrt{(-1)^{2} + (-4)^{2}} = \sqrt{1 + 16} = \sqrt{17}$.
    • Perimeter: $4 + 7 + \sqrt{20} + \sqrt{17} \approx 19.6\,units$.

1-7: Transformations in the Plane

• Application: Fashion Design Prints

  • Patterns are created by sliding (translation), flipping (reflection), or turning (rotation) figures.

• Types of Rigid Transformations

  • Rigid Transformation: A transformation that preserves the size and shape of a figure, resulting in a congruent image.
  • Translation: Sliding a figure to a different location. It maintains the orientation of the figure.
  • Reflection: Flipping a figure over a line to create a mirror image.
  • Rotation: Turning a figure around a fixed point.

• Coordinate Rules for Transformations

  • Reflection across x-axis: $(a, b) \rightarrow (a, -b)$.
  • Reflection across y-axis: $(a, b) \rightarrow (-a, b)$.
  • Translation: If translated along vector $\langle x, y \rangle$, then $(a, b) \rightarrow (a + x, b + y)$.
  • Rotations about the Origin:
    • $90^{\circ}$ Rotation: $(x, y) \rightarrow (-y, x)$.
    • $180^{\circ}$ Rotation: $(x, y) \rightarrow (-x, -y)$.
    • $270^{\circ}$ Rotation: $(x, y) \rightarrow (y, -x)$.

• Practice Examples: Transformations

  • Reflection in x-axis: $\Delta PSR$ with $P(-3, 4), S(-3, -1), R(-6, 2)$ becomes $P'(-3, -4), S'(-3, 1), R'(-6, -2)$.
  • Translation along Vector $\langle -1, 2 \rangle$: Parallelogram $ABCD$ with $A(-2, 1), B(-1, 4), C(3, 1), D(4, 4)$ becomes $A'(-3, 3), B'(-2, 6), C'(2, 3), D'(3, 6)$.
  • $180^{\circ}$ Rotation: Quadrilateral $ABCD$ with $A(-4, -2), B(-3, -3), C(-2, -1), D(-1, 0)$ becomes $A'(4, 2), B'(3, 3), C'(2, 1), D'(1, 0)$.

1-8: Surface Area and Volume of Three-Dimensional Figures

• Polyhedrons

  • Definition: A solid figure made up of flat surfaces (polygons) that enclose a region of space.
  • Examples: Prisms and Pyramids.
  • Non-Polyhedrons: Cylinders, Cones, and Spheres (contain curved surfaces).
  • Parts of a Polyhedron:
    • Face: A flat surface.
    • Edge: The line segment where two faces meet.
    • Vertex: The point where three or more edges intersect.
    • Base: The faces used to name the solid.

• Surface Area (S or SA) Formulas

  • Prism: $S = Ph + 2B$
  • Pyramid: $S = \frac{1}{2}Pl + B$
  • Cylinder: $S = Ch + 2B = 2\pi rh + 2\pi r^{2}$
  • Cone: $S = \pi rl + B = \pi rl + \pi r^{2}$
  • Sphere: $S = 4\pi r^{2}$
  • Hemisphere: $S = \frac{1}{2}(4\pi r^{2}) + \pi r^{2} = 3\pi r^{2}$
  • Variables:
    • $B$: Area of the base.
    • $P$: Perimeter of the base.
    • $C$: Circumference of the base.
    • $h$: Height of the solid.
    • $l$: Slant height (distance from the vertex to the edge of the base along the surface curved/slanted side).
    • Watch Out: Height ($h$) is the vertical distance to the base; slant height ($l$) is different.

• Volume (V) Formulas

  • Volume is measured in cubic units ($units^{3}$).
  • Prism: $V = Bh$
  • Cylinder: $V = Bh = \pi r^{2}h$
  • Pyramid: $V = \frac{1}{3}Bh$
  • Cone: $V = \frac{1}{3}Bh = \frac{1}{3}\pi r^{2}h$
  • Sphere: $V = \frac{4}{3}\pi r^{3}$
  • Hemisphere: $V = \frac{2}{3}\pi r^{3}$

• Specific Examples: Surface Area and Volume

  • Rectangular Prism: Base sides $8\,in$ and $12\,in$, height $6\,in$.
    • $S = (40)(6) + 2(96) = 240 + 192 = 432\,in^{2}$.
    • $V = (96)(6) = 576\,in^{3}$.
  • Cylinder: Radius $4\,cm$, height $9\,cm$.
    • $S = 2\pi(4)(9) + 2\pi(4^{2}) = 72\pi + 32\pi = 104\pi \approx 326.73\,cm^{2}$.
    • $V = \pi(4^{2})(9) = 144\pi \approx 452.39\,cm^{3}$.
  • Sphere: Radius $5\,ft$.
    • $V = \frac{4}{3}\pi(5^{3}) = \frac{500}{3}\pi \approx 523.6\,ft^{3}$.
  • Hemisphere: Radius $10\,ft$.
    • Volume: $V = \frac{2}{3}\pi(10^{3}) \approx 2094.4\,ft^{3}$.
  • Cone (Icing Bag):
    • Diameter $3.5\,in$ ($r = 1.75\,in$), height $5\,in$, slant height $5.3\,in$.
    • Volume: $V = \frac{1}{3}\pi(1.75^{2})(5) \approx 16.0\,in^{3}$.
    • Surface Area (no top): $S = \pi(1.75)(5.3) \approx 29.1\,in^{2}$.

Questions & Discussion

• Q: What geometric figure comprises the sides of a polygon?

  • A: Line segments.

• Q: How do you name a polygon?

  • A: Based on the number of sides of the polygon.

• Q: What is the name for a convex polygon that is equilateral and equiangular?

  • A: A regular polygon.

• Q: How do you label units for perimeter or circumference? Area?

  • A: Perimeter and circumference use linear units (e.g., $ft, cm$). Area uses square units (e.g., $ft^{2}, cm^{2}$).

• Q: Which transformation maintains the orientation of a figure?

  • A: Translation.

• Q: Why do rigid transformations create congruent figures?

  • A: Because they do not change the shape or size of the original figure.

• Q: Which three-dimensional solids are polyhedrons? Which are not?

  • A: Pyramids and prisms are polyhedrons. Spheres, cylinders, and cones are not polyhedrons.

• Q: How do you label units for volume?

  • A: Volume is labeled with cubic units (e.g., $ft^{3}, cm^{3}$).

• Quiz Correction Logic for Vanesa's Banner:

  • The problem asks which shape uses most of $20\,sq\,ft$ of fabric.
  • Option C ($r = 2.5\,ft$ circle) gives $A = \pi(2.5^{2}) = 19.63\,ft^{2}$, which is the closest value under $20$ without exceeding it.