Basic Geometric Formulas

Understanding basic geometric formulas is key in Elementary Algebraic Geometry. These formulas help calculate areas, volumes, and distances, forming the foundation for more complex concepts. Mastering them will enhance your problem-solving skills in geometry and beyond.

1. Area of a rectangle: A = length × width

   • The area represents the total space within the rectangle.
   • Both length and width must be in the same unit for accurate calculation.
   • The formula can be applied to any rectangle, regardless of size.

2. Area of a triangle: A = (base × height) / 2

   • The area is half the product of the base and height.
   • The height must be perpendicular to the base for the formula to be valid.
   • This formula applies to all types of triangles (scalene, isosceles, equilateral).

3. Area of a circle: A = πr²

   • The area is determined by the radius squared multiplied by π (approximately 3.14).
   • The radius is the distance from the center to any point on the circle.
   • This formula is essential for calculating space in circular shapes.

4. Circumference of a circle: C = 2πr

   • The circumference is the total distance around the circle.
   • It is calculated by multiplying the diameter (2r) by π.
   • Understanding circumference is crucial for problems involving circular motion.

5. Pythagorean theorem: a² + b² = c²

   • This theorem relates the lengths of the sides of a right triangle.
   • 'c' represents the hypotenuse, while 'a' and 'b' are the other two sides.
   • It is fundamental in various applications, including distance and geometry.

6. Volume of a rectangular prism: V = length × width × height

   • The volume measures the space inside the prism.
   • All dimensions must be in the same unit for accurate results.
   • This formula is applicable to any rectangular box-like shape.

7. Volume of a cylinder: V = πr²h

   • The volume is calculated by multiplying the area of the base (circle) by the height.
   • The radius must be squared and then multiplied by π and the height.
   • This formula is essential for understanding three-dimensional circular shapes.

8. Surface area of a sphere: SA = 4πr²

   • The surface area measures the total area covering the sphere.
   • It is calculated using the radius squared multiplied by 4π.
   • This formula is important in fields involving spherical objects.

9. Volume of a sphere: V = (4/3)πr³

   • The volume represents the space within the sphere.
   • The radius is cubed and then multiplied by (4/3)π.
   • This formula is crucial for calculations involving spherical volumes.

10. Distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]

    • The distance formula calculates the straight-line distance between two points in a plane.
    • It uses the coordinates of the points (x₁, y₁) and (x₂, y₂).
    • This formula is fundamental in geometry and coordinate systems.