Basic Geometry
Shapes, areas, perimeters, and volumes — the math of the physical world around you.
Basic Shapes and Their Properties
Angles
- Acute: Less than 90°
- Right: Exactly 90° (the corner of a square)
- Obtuse: Between 90° and 180°
- Straight: Exactly 180° (a flat line)
Angles in a triangle always add up to 180°. Angles in a quadrilateral add up to 360°.
Perimeter (Distance Around)
Perimeter = total distance around the outside of a shape.
Rectangle
P = 2(length + width)
Example 1: A room 15 ft × 10 ft. P = 2(15+10) = 2(25) = 50 ft
Square
P = 4 × side
Example 2: A 6-inch square. P = 4 × 6 = 24 inches
Triangle
P = side₁ + side₂ + side₃
Example 3: Sides 5, 7, 9. P = 5 + 7 + 9 = 21
Circle (Circumference)
C = 2πr = πd (where r = radius, d = diameter, π ≈ 3.14159)
Example 4: Diameter 10 inches. C = π × 10 ≈ 31.4 inches
Area (Space Inside)
Rectangle
A = length × width
Example 5: A 12×8 room. A = 96 sq ft
Triangle
A = ½ × base × height
Example 6: Base 14, height 8. A = ½ × 14 × 8 = 56 sq units
Circle
A = πr²
Example 7: Radius 5. A = π × 5² = π × 25 ≈ 78.5 sq units
Trapezoid
A = ½ × (base₁ + base₂) × height
Example: Bases 6 and 10, height 4. A = ½(6+10) × 4 = ½ × 16 × 4 = 32
Volume (3D Space)
Rectangular Box (Prism)
V = length × width × height
Example 8: A box 10 × 6 × 4. V = 240 cubic units
Cylinder
V = πr²h
Example: Radius 3, height 10. V = π × 9 × 10 ≈ 282.7 cubic units
Sphere
V = (4/3)πr³
Example: Radius 6. V = (4/3) × π × 216 ≈ 904.8 cubic units
The Pythagorean Theorem
For any right triangle: a² + b² = c²
Where c is the hypotenuse (longest side, opposite the right angle).
Example 9: Find the missing side
Legs: a = 6, b = 8. Find c.
6² + 8² = c² → 36 + 64 = 100 → c = √100 = 10
Common Pythagorean Triples
3-4-5, 5-12-13, 8-15-17, 7-24-25
Any multiple works too: 6-8-10, 9-12-15, etc.
Real-World Applications
- Flooring: Area tells you how many sq ft of hardwood to buy
- Fencing: Perimeter tells you how much fence for your yard
- Concrete: Volume tells you how many cubic yards to order
- Diagonal measurement: Pythagorean theorem finds TV screen sizes
🎯 Try It Yourself
Test your understanding with these practice problems.
1. What is the area of a rectangle that is 12 ft long and 8 ft wide?
💡 Hint: Area = length × width
2. What is the circumference of a circle with radius 7? (Use π ≈ 3.14, round to nearest tenth)
💡 Hint: C = 2πr = 2 × 3.14 × 7
3. What is the area of a triangle with base 10 and height 6?
💡 Hint: Area = ½ × base × height
4. What is the volume of a box 5 × 4 × 3?
💡 Hint: Volume = length × width × height
5. A right triangle has legs of 3 and 4. What is the hypotenuse?
💡 Hint: a² + b² = c². So 9 + 16 = 25. √25 = ?
⚠️ Common Mistakes
- ✗Confusing area (square units) with perimeter (linear units). Area fills the inside; perimeter goes around the edge.
- ✗Using diameter when the formula calls for radius, or vice versa. Radius = diameter ÷ 2.
- ✗Forgetting to square the units for area (sq ft, m²) and cube them for volume (cu ft, m³).
- ✗Using the wrong formula for triangles — area is ½bh, NOT bh. The ½ is essential!
- ✗Applying the Pythagorean theorem to non-right triangles. It only works when there's a 90° angle.
🌍 Real Life Example
Painting a Room
You're painting a room that is 14 ft × 12 ft with 8 ft ceilings. Wall area: 2(14×8) + 2(12×8) = 224 + 192 = 416 sq ft. Subtract for a door (21 sq ft) and 2 windows (2 × 15 = 30 sq ft): 416 - 21 - 30 = 365 sq ft to paint. One gallon covers ~350 sq ft, so you need 2 gallons. Geometry literally saves you money by buying the right amount of paint.
💡 Key Takeaway
Geometry connects math to the real world. Key formulas: Rectangle area = l×w, Triangle area = ½bh, Circle area = πr². Perimeter is the distance around, area is the space inside, volume is the space a 3D object fills. The Pythagorean theorem (a²+b²=c²) works for all right triangles.