Overview
This lecture reviews essential geometry concepts, formulas, and problem-solving strategies for circles, squares, rectangles, and right triangles, focusing on key equations and special triangle relationships for standardized tests.
Circles
- Circumference formula: (C = 2\pi r)
- Area formula: (A = \pi r^2)
- Diameter formula: (d = 2r)
- Chord: a line connecting two points on the circle, not passing through the center
- Example: For (r = 5), (C = 10\pi \approx 31.42), (A = 25\pi \approx 78.54), (d = 10)
Squares
- Area formula: (A = s^2) (where (s) is the side length)
- Perimeter formula: (P = 4s)
- Example: (s = 8), (A = 64), (P = 32)
- To find side from area, use (s = \sqrt{A})
Rectangles
- Area formula: (A = l \times w) (length times width)
- Perimeter formula: (P = 2l + 2w)
- To find width from area and length: (w = A / l)
- Given relationships (e.g., length is three more than twice the width), set up equations and solve
- Use substitution and factoring for quadratic equations in word problems
Right Triangles & Pythagorean Theorem
- Formula: (a^2 + b^2 = c^2) (c is hypotenuse)
- Know these special triangles: 3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-40-41, 11-60-61
- Multiples of special triangles are also right triangles (e.g., 6-8-10 is double 3-4-5)
- Triangle side relationships help solve for missing sides quickly without calculations
Problem-Solving Strategies
- Use substitution for equations involving unknowns
- Factor trinomials for quadratic equations in geometry problems
- When given area or perimeter with variable sides, set up equations using geometric formulas
Key Terms & Definitions
- Circumference — The distance around a circle ((2\pi r))
- Area — The space inside a shape (e.g., circle: (\pi r^2), square: (s^2), rectangle: (l \times w))
- Diameter — Distance across a circle through its center ((2r))
- Chord — Line segment joining two points on a circle, not passing through the center
- Perimeter — The total length around a shape
Action Items / Next Steps
- Practice using the key geometry formulas for circles, squares, rectangles, and triangles
- Memorize common right triangle side ratios (special triangles)
- Solve sample SAT/ACT and practice problems involving these geometric concepts and formulas