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Circles & Area

The perfect shape — and the formulas for measuring every kind of region.

Circle Basics

A circle is the set of all points equidistant from a center point. This distance is the radius r.

Circumference: C = 2πr
Area: A = πr²
Equation: (x − h)² + (y − k)² = r²   (center (h,k), radius r)

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.

Related Topics

The Unit Circle The circle of radius 1 that defines all trigonometric functions. Integrals Compute areas, volumes, and arc lengths using integration. Coordinate Geometry Circle equations and conic sections on the coordinate plane. Formula Sheet Quick reference for all area, volume, and surface area formulas.