geom 3.06

Unit 3: Measurement of Two-Dimensional Figures

Lesson 1

• Altitude: A perpendicular line segment that measures the height of a geometric figure.

• Area: The number of square units contained in the interior of a figure.

• Composite Figure: A figure made up of two or more figures.

• Perimeter: The distance around a plane figure.

Formulas

• Distance Formula: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

• Area of a Rectangle: [ A = l \times w ]

• Area of a Triangle: [ A = \frac{1}{2} b \times h ]

Steps to Find Area of Composite Figures

1. Divide shape into non-overlapping triangles and rectangles.

2. Find area of each polygon.

3. Add areas.

Lesson 2

• Apothem: A line segment from the center of a regular polygon to a side, perpendicular to that side.

Area of a Regular Polygon

• Formula: [ A = \frac{1}{2} P a ]

  • Where ( A ) is area, ( P ) is perimeter, and ( a ) is apothem.

Example: Area of Regular Hexagon

1. Find perimeter

2. Plug values into formula

3. Multiply.

Lessons 3 & 4

• Circle: The set of all points equidistant from a center point.

• Circumference: The distance around a circle.

Circumference Formula

• Formula: [ C = \pi d ]

  • Example with Ferris wheel radius 80 ft: [ C = 3.14 \times 80 = 251.2 , \text{feet} ]

Lessons 6 & 7

Area Formulas

• Area of a Semicircle: [ A = \frac{1}{2} \pi r^2 ]

• Area of a Rectangle: List dimensions for area calculation.

• Area of a Triangle: [ A = \frac{1}{2} b h ]

Total Area of Composite Figure:

• Calculate area of all shapes combined.

Lesson 9

Arc Length Formulas

• If Intercepted Arc is in Degrees: [ s = \frac{\theta}{360} \times 2\pi r ]

• If Intercepted Arc is in Radians: [ s = r\theta ]

Lesson 10

• Sector: A region in a circle defined by two radii and the arc between them.

Area of a Sector Formulas

• Formula for Degrees: [ A = \frac{\theta}{360} \times \pi r^2 ]

• Example: Area of a 30° sector.

• Sector Area Calculation: Use radius, central angle, and plug into the formula.