In This Lesson Circle Basics Arcs, Sectors & Segments Inscribed Angles Area Formulas Volume & Surface Area Circle Basics 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 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)
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 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
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 Inscribed Angles 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 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 .
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 A comprehensive reference for all 2D shapes (also on the formula sheet ):
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)
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 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.
vA4a4S18ZQ4CRNdKpI3Ln25Mr2dkOuS2J8LlGL90ovdOkmVOQpg0NoLEiVXBz7FeiRvytgG+2BO5Z33TLh+5dNANyfFvmY7kbrc+zfGRBSnhVZsQNwCXDz3CY7weSo5R8K/icwhK7I/OLxFrZj3Cijp8KzD8PKCTet1oCfM92n9M5Ey+57lyeE/tcLKwKAh11EEe7XdKE/hwLXQNDL1TFnCLWULHfkOJ6Dohgegn5ooCaKTZK4ieGRoOG1RVYmCEvvBC0TqJWClR6dcETwZsSWdqJ62cn16tgK6jlcV0TwPDbkyaHI75kXk/9voiv7VIGjxNpcwlBAnQhgcruj90FVx0ho22MyXo4kmnpJQ/RiKpNqIsT097j3o10z0ypxyYPlUSUGQpovdut77CwiPxqFThE8dKwiJjUrja76rnE6AqAiqQXWeQt9nTOP269qsKdQWTvuheeOv69WYTID7R2XkqM4LGsZLXF/tYOHEzi4EUOJ6L+zdLtG2OGdkavDC/EPwHsqoUHNQkQuhdQfDJG9fFdPz0vypdv49h/VE3y3JzCvea3yN45bv09qCRrAUtOT76e3t3FZPJHJs/bHQaJaCflxXb0peyhplLp86M/OYY3bTjC+Yh1orN+k1wWeUbuX+cQtsIFhIE4V1mi+HLEydRpJyTaQIiOlkigg