📖 What is Geometry?
Geometry is the branch of mathematics that studies shapes, sizes, positions, and properties of space. From ancient Egyptian pyramids to modern architecture, geometry is everywhere in our world.
Why Learn Geometry?
- Spatial Reasoning: Develops ability to visualize and manipulate shapes mentally
- Logical Thinking: Proofs teach structured reasoning and argumentation
- Real Applications: Used in architecture, engineering, art, and design
- Foundation for Advanced Math: Essential for trigonometry, calculus, and physics
🎯 Key Concepts
1. Basic Building Blocks
Fundamental Definitions
- Point: A location in space with no size (represented by a dot)
- Line: Infinite set of points extending in both directions
- Line Segment: Part of a line with two endpoints
- Ray: Part of a line with one endpoint, extending infinitely in one direction
- Angle: Formed by two rays sharing a common endpoint (vertex)
- Plane: A flat surface extending infinitely in all directions
2. Angles
Types of Angles
- Acute: Less than 90°
- Right: Exactly 90°
- Obtuse: Between 90° and 180°
- Straight: Exactly 180°
- Reflex: Between 180° and 360°
Angle Relationships
- Complementary: Two angles that sum to 90°
- Supplementary: Two angles that sum to 180°
- Vertical Angles: Opposite angles formed by intersecting lines (always equal)
- Adjacent Angles: Angles that share a common side and vertex
3. Triangles
Triangles are the simplest polygons and form the basis for understanding more complex shapes.
Types by Sides
- Equilateral: All three sides equal (all angles = 60°)
- Isosceles: Two sides equal (base angles are equal)
- Scalene: No sides equal (no angles equal)
Types by Angles
- Acute: All angles less than 90°
- Right: One angle equals 90°
- Obtuse: One angle greater than 90°
For right triangles: $a$ and $b$ are legs, $c$ is the hypotenuse
4. Quadrilaterals
Four-sided polygons with various special properties.
5. Circles
Circle Vocabulary
- Radius ($r$): Distance from center to any point on circle
- Diameter ($d$): Distance across circle through center ($d = 2r$)
- Chord: Line segment with both endpoints on the circle
- Arc: Part of the circumference
- Sector: "Pie slice" region between two radii
- Tangent: Line that touches circle at exactly one point
6. Congruence & Similarity
Triangle Congruence Postulates
- SSS: Side-Side-Side (all three sides equal)
- SAS: Side-Angle-Side (two sides and included angle)
- ASA: Angle-Side-Angle (two angles and included side)
- AAS: Angle-Angle-Side (two angles and non-included side)
- HL: Hypotenuse-Leg (for right triangles only)
Similar Triangles
Triangles are similar if they have the same shape but different sizes. All corresponding angles are equal, and corresponding sides are proportional.
- AA: Two angles equal (third is automatically equal)
- SSS: All sides proportional
- SAS: Two sides proportional with equal included angle
📝 Essential Formulas
Area Formulas
| Shape | Area Formula | Variables |
|---|---|---|
| Triangle | $A = \frac{1}{2}bh$ | $b$ = base, $h$ = height |
| Rectangle | $A = lw$ | $l$ = length, $w$ = width |
| Square | $A = s^2$ | $s$ = side length |
| Parallelogram | $A = bh$ | $b$ = base, $h$ = height |
| Trapezoid | $A = \frac{1}{2}(b_1 + b_2)h$ | $b_1, b_2$ = parallel bases |
| Circle | $A = \pi r^2$ | $r$ = radius |
| Sector | $A = \frac{\theta}{360°} \pi r^2$ | $\theta$ = central angle |
Perimeter & Circumference
| Shape | Formula |
|---|---|
| Rectangle | $P = 2l + 2w$ |
| Square | $P = 4s$ |
| Triangle | $P = a + b + c$ |
| Circle (Circumference) | $C = 2\pi r = \pi d$ |
| Arc Length | $L = \frac{\theta}{360°} \times 2\pi r$ |
3D Shapes
| Shape | Volume | Surface Area |
|---|---|---|
| Cube | $V = s^3$ | $SA = 6s^2$ |
| Rectangular Prism | $V = lwh$ | $SA = 2(lw + lh + wh)$ |
| Cylinder | $V = \pi r^2 h$ | $SA = 2\pi r^2 + 2\pi rh$ |
| Sphere | $V = \frac{4}{3}\pi r^3$ | $SA = 4\pi r^2$ |
| Cone | $V = \frac{1}{3}\pi r^2 h$ | $SA = \pi r^2 + \pi r l$ |
| Pyramid | $V = \frac{1}{3}Bh$ | $SA = B + \frac{1}{2}Pl$ |
Special Right Triangles
If leg = $x$, then hypotenuse = $x\sqrt{2}$
Short leg = $x$, long leg = $x\sqrt{3}$, hypotenuse = $2x$
✏️ Practice Problems
$A = \frac{1}{2}bh$
$A = \frac{1}{2}(12)(8)$
$A = \frac{1}{2}(96) = 48$
Answer: 48 cm²
$a^2 + b^2 = c^2$
$5^2 + 12^2 = c^2$
$25 + 144 = c^2$
$169 = c^2$
$c = \sqrt{169} = 13$
Answer: 13
$C = 2\pi r = 2(3.14)(7) = 43.96$ cm
$A = \pi r^2 = (3.14)(7)^2 = (3.14)(49) = 153.86$ cm²
Answer: C = 43.96 cm, A = 153.86 cm²
30-60-90 triangles have sides in ratio $1 : \sqrt{3} : 2$
This corresponds to the "1" in the ratio
Long leg (opposite 60°) = $6\sqrt{3} \approx 10.39$
Hypotenuse (opposite 90°) = $6 \times 2 = 12$
Answer: $6\sqrt{3}$ and 12
$V = \pi r^2 h$
$V = \pi (4)^2 (10)$
$V = \pi (16)(10)$
$V = 160\pi$
$V \approx 160(3.14) = 502.4$ cm³
Answer: $160\pi$ cm³ ≈ 502.4 cm³
💡 Tips & Tricks
Always sketch the problem! Label all given information and what you're trying to find. Visual representation makes problems much clearer.
Know 45-45-90 ($1:1:\sqrt{2}$) and 30-60-90 ($1:\sqrt{3}:2$) ratios by heart. They appear constantly!
Complex figures often contain triangles, rectangles, or circles. Break them into simpler shapes you know formulas for.
Memorize common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Multiples work too (6-8-10, 9-12-15)!
In pyramids and cones, height goes straight down from apex to base center. Slant height goes along the surface. Don't confuse them!
For proofs: (1) List what's given, (2) State what to prove, (3) Work backward from conclusion while working forward from given facts until they meet.
⚠️ Common Mistakes to Avoid
Remember: Perimeter is the distance AROUND a shape (measured in units). Area is the space INSIDE (measured in square units).
Circle formulas use radius! If given diameter, divide by 2 first.
$C = 2\pi r$ (not $2\pi d$) and $A = \pi r^2$ (not $\pi d^2$)
The height must be PERPENDICULAR to the base. It's not always a side of the triangle—sometimes you need to draw it!
Always include units! Area is in square units (cm², m²), volume is in cubic units (cm³, m³).
Shapes that LOOK equal might not be. Only use what's explicitly stated or proven. Don't assume angles are 90° unless marked!
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