Key Definitions
- Apothem:
-
A line from the center of a regular polygon to the midpoint of one of its sides, forming a right angle with the side.
-
Axis:
-
A reference line for measurement or symmetry about which an object can be rotated or divided.
-
Radius:
- The distance from the center of a circle or sphere to its edge (circumference for a circle, surface for a sphere).
Quick Reference for Formulas
2D Shapes (Area)
- Square:
-
( A = {side}^2 )
-
Rectangle:
-
( A = {length} * {width} )
-
Triangle:
-
( A = \frac{{base} * {height}}{2} )
-
Circle:
-
( A = \pi r^2 )
-
Parallelogram:
-
( A = {base} * {height} )
-
Trapezium (Trapezoid):
- ( A = \frac{1}{2} ({base}_1 + {base}_2) * {height} )
3D Shapes (Surface Area and Volume)
- Cube:
- Surface Area: ( 6 * {side}^2 )
-
Volume: ( {side}^3 )
-
Rectangular Prism (Cuboid):
- Surface Area: ( 2(lw + lh + wh) )
-
Volume: ( l * w * h )
-
Sphere:
- Surface Area: ( 4\pi r^2 )
-
Volume: ( \frac{4}{3} \pi r^3 )
-
Cylinder:
- Surface Area: ( 2\pi r(h + r) )
-
Volume: ( \pi r^2 h )
-
Cone:
- Surface Area: ( \pi r(r + l) ) (where ( l ) is the slant height)
-
Volume: ( \frac{1}{3} \pi r^2 h )
-
Pyramid:
- Surface Area: Base Area + ( \frac{1}{2} * {Perimeter} * {Slant Height} )
- Volume: ( \frac{1}{3} * {Base Area} * {Height} )
Tips for Applying These Formulas
- Ensure measurements are in the same units before calculating.
- For composite shapes, break them into smaller, manageable parts (e.g., rectangles, triangles).
- Use ( \pi = 3.142 ) or the exact value on your calculator for better precision in circle and sphere calculations.
- Double-check calculations for complex shapes, especially those involving multiple dimensions or components (e.g., pyramids, cones).
This reference sheet is a helpful guide for quick geometry-related calculations!