Common Circle formulas with brief explanations. These formulas are useful for solving problems involving circles in math, geometry and trigonometry.
Circle formulas
Circumference (Perimeter):
- Formula: C = 2πr or C = πd
- Explanation: The circumference of a circle (C) is the total length of its boundary. It can be calculated using either the radius (r) or the diameter (d) of the circle.
Area:
- Formula: A = πr²
- Explanation: The area of a circle (A) is the amount of space enclosed by its boundary. It is calculated using the square of the radius (r) and the constant π (pi).
Diameter:
- Formula: d = 2r
- Explanation: The diameter of a circle (d) is the distance across the circle through its center. It is always twice the length of the radius (r).
Radius:
- Formula: r = d/2
- Explanation: The radius of a circle (r) is the distance from its center to any point on its boundary. It is half of the diameter (d).
Arc Length:
- Formula: L = (θ/360) * 2πr
- Explanation: The arc length (L) is the length of a portion of the circle's circumference, determined by a central angle (θ) measured in degrees. This formula calculates the length of an arc in a circle.
- Formula: L = (θ/360) * 2πr
Sector Area:
- Formula: A = (θ/360) * πr²
- Explanation: The sector area (A) is the area of a portion of the circle, determined by a central angle (θ) measured in degrees. This formula calculates the area of a sector in a circle.
Chord Length:
- Formula: L = 2r * sin(θ/2)
- Explanation: The chord length (L) is the distance between two points on the circle's circumference, determined by a central angle (θ) measured in degrees. This formula calculates the length of a chord in a circle using the radius (r) and the sine of half the central angle.
Tangent Line Length:
- Formula: L = √(r² - x²)
- Explanation: The length of a tangent line (L) from an external point (x) to the circle is calculated using the radius (r) and the distance from the point to the circle's center. This formula is derived from the Pythagorean theorem.
Sector Angle:
- Formula: θ = (A/A₀) * 360°
- Explanation: The sector angle (θ) is the central angle corresponding to a sector's area (A) in a circle. A₀ represents the total area of the circle (πr²), and this formula helps calculate the central angle for a given sector area.
Inscribed Angle Theorem:
- Formula: θ = 0.5 * m(arc)
- Explanation: In the context of a circle, the inscribed angle (θ) formed by two chords is half the measure of the intercepted arc (arc). This theorem is useful for solving problems involving inscribed angles.
Chord Midpoint Formula:
- Formula: h² = r² - (0.5c)²
- Explanation: Given a chord's length (c) and the circle's radius (r), this formula allows you to calculate the distance (h) from the chord's midpoint to the center of the circle.
Segment Height (Altitude):
- Formula: h = r - √(r² - d²)
- Explanation: This formula calculates the height (h) of a circle segment, which is the perpendicular distance from the chord's midpoint to the circle's arc. It requires the radius (r) and the chord's length (d).
Tangent Line and Radius Relation:
- Formula: tan(θ) = h/r
- Explanation: In a circle, this formula relates the tangent of the angle (θ) formed between a radius and a tangent line to the circle to the height of the segment (h) and the radius (r). It's useful for solving problems involving tangents.
Heron's Formula for the Area of a Triangle Inscribed in a Circle:
- Formula: A = √[s(s - a)(s - b)(s - c)]
- Explanation: This formula calculates the area (A) of a triangle inscribed in a circle, where "a," "b," and "c" are the lengths of the triangle's sides, and "s" is the semi-perimeter of the triangle (s = (a + b + c)/2). It's useful when dealing with cyclic triangles.
Euler's Formula for the Relationship Between Vertices, Edges, and Faces in a Convex Polyhedron Inscribed in a Sphere:
- Formula: V - E + F = 2
- Explanation: Euler's formula relates the number of vertices (V), edges (E), and faces (F) of a convex polyhedron inscribed in a sphere. It states that for any such polyhedron, the sum of vertices and faces minus the number of edges is always equal to 2.
Circle Calculation Examples
Circle examples with practical calculations for each of the above 15 circle related formulas.
Circumference (Perimeter):
- Given radius (r) = 2 inches
- C = 2πr = 2π(2) ≈ 12.57 inches
Area:
- Given radius (r) = 3 inches
- A = πr² = π(3²) ≈ 28.27 square inches
Diameter:
- Given radius (r) = 4 inches
- d = 2r = 2(4) = 8 inches
Radius:
- Given diameter (d) = 10 inches
- r = d/2 = 10/2 = 5 inches
Arc Length:
- Given central angle (θ) = 45 degrees and radius (r) = 6 inches
- L = (45/360) * 2π(6) ≈ 4.71 inches
Sector Area:
- Given central angle (θ) = 60 degrees and radius (r) = 7 inches
- A = (60/360) * π(7²) ≈ 22.00 square inches
Chord Length:
- Given radius (r) = 8 inches and central angle (θ) = 30 degrees
- L = 2(8) * sin(30/2) ≈ 7.79 inches
Tangent Line Length:
- Given radius (r) = 5 inches and distance (x) = 3 inches
- L = √(5² - 3²) = √(25 - 9) = √16 = 4 inches
Sector Angle:
- Given sector area (A) = 10 square inches and total area (A₀) = 16 square inches
- θ = (10/16) * 360° ≈ 225 degrees
Inscribed Angle Theorem:
- Given intercepted arc (arc) measure = 120 degrees
- θ = 0.5 * 120 = 60 degrees
Chord Midpoint Formula:
- Given chord length (c) = 9 inches and radius (r) = 6 inches
- h² = 6² - (0.5 * 9)² = 36 - 20.25 = 15.75
- h ≈ √15.75 ≈ 3.98 inches
Segment Height (Altitude):
- Given radius (r) = 10 inches and chord length (d) = 8 inches
- h = 10 - √(10² - 8²) ≈ 6 inches
Tangent Line and Radius Relation:
- Given height (h) = 3 inches and radius (r) = 7 inches
- tan(θ) = 3/7 ≈ 0.43
Heron's Formula for the Area of a Triangle Inscribed in a Circle:
- Given side lengths: a = 4 inches, b = 6 inches, c = 7 inches
- Calculate the semi-perimeter (s) first.
- s = (4 + 6 + 7)/2 = 8.5 inches
- A = √[8.5(8.5 - 4)(8.5 - 6)(8.5 - 7)] ≈ 14.70 square inches
Euler's Formula for Convex Polyhedron Inscribed in a Sphere:
- For a cube: V = 8, E = 12, F = 6
- Euler's formula: V - E + F = 8 - 12 + 6 = 2 (as expected)