The circle equation is a conic section — one of the four curves obtained by slicing a cone. The constant π ≈ 3.14159 appears throughout mathematics; see the formula sheet for its properties.
Arcs, Sectors & Segments
Arc length: s = rθ (θ in radians)
Sector area: A = ½r²θ
Segment area: A = ½r²(θ − sin θ)
These formulas use radians — the natural angle measure for trigonometry and calculus. One full revolution = 2π radians = 360°.
Example: Arc length for θ = π/3 on r = 6
s = 6 · π/3 = 2π ≈ 6.28
Inscribed Angles
An angle inscribed in a circle (vertex on the circumference) is half the central angle that subtends the same arc:
Inscribed angle = ½ × central angle
Special case: An inscribed angle that subtends a semicircle is always 90° — Thales' theorem. This ancient result connects circles to right triangles.
Area Formulas
A comprehensive reference for all 2D shapes (also on the formula sheet):
Rectangle: A = lw
Triangle: A = ½bh or A = ½ab·sin(C) — uses trig
Parallelogram: A = bh
Trapezoid: A = ½(b₁ + b₂)h
Regular polygon (n sides, side s): A = (ns²)/(4·tan(π/n))
Circle: A = πr²
Ellipse: A = πab
The area of a circle can be derived using integration: A = ∫₋ᵣ ʳ 2√(r² − x²) dx = πr². This is one of the first applications of integral calculus to geometry.
Volume & Surface Area
Rectangular prism: V = lwh, SA = 2(lw + lh + wh)
Cylinder: V = πr²h, SA = 2πrh + 2πr²
Cone: V = ⅓πr²h, SA = πr√(r² + h²) + πr²
Sphere: V = (4/3)πr³, SA = 4πr²
Pyramid: V = ⅓Bh (B = base area)
The volume formulas for cones and pyramids (with the ⅓ factor) can be proved rigorously using integral calculus. Archimedes originally derived the sphere volume using a brilliant geometric argument — one of the greatest achievements of ancient mathematics.
In linear algebra, the determinant of a matrix gives the volume scaling factor of the associated linear transformation. A 3×3 matrix with determinant 2 doubles all volumes. This connects geometric volume to algebraic structure.