In This Lesson Circle Basics Arcs, Sectors & Segments 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 Inscribed Angles Area Formulas 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 Volume & Surface Area 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 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
Inscribed Angles An angle inscribed in a circle (vertex on the circumference) is half the central angle that subtends the same arc:
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 Inscribed angle = ½ × central angle
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 Special case: An inscribed angle that subtends a semicircle is always 90° — Thales' theorem . This ancient result connects circles to right triangles .
A comprehensive reference for all 2D shapes (also on the formula sheet ):
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
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.
+kYdrCi1O0VLo5yamH/YU4kfEjs6Lmh+imheOaQA0fGIqb6fRpq+GGpJmKIBGTFWLUYAST6Ifoq/bzygIIbAvTWyGX3FH3bbwn7Dm6Le3i8MCbbJLj++RRPQluMXyArhDYQaCETQwr46YDRrCKI3o/PuhzUbovXTZoZWJa1nkekfo0lS2fvx0YuWhTTesqvWefp8RX4sh2AvXTk3NwnitNdF9pnvBhrtYg+iXNgyr5cZlap7WnPGL5goquSZ0SbHw29WnrWvmkxf2e6aN+565W2H/B8+kwZePi2oMgGc0d98ryQxmZSkQGURCmgUfY4HcXp7Y90vLFyB7Amj/sRsI5nVf1LxtPe5cvipAGG5LA3xxUZWc7VPVYOhIFPEUtfOIF0I0shNUBuCWpyZ4TDcccXzbi6fQ53geBStGFMrIP+PhB8TZIu3xS8rgoH4EtI0kZRZrzHtyiho5FOT2RImDUxNVWKus4jzfks9v3uq8kLKV1Z4vdKvOKXQp+Ut7dTbn1+uaFbz9zdq7/nZ/vPpyYHXILvymWw8S2n3OY2RIQGvZDA2jt+eNFqovSSg1CPT59kWhanfLDeshmJxT0FU0ipNs6tNtjdhRwMI0jeLzWuDuoZJsgdsg9hp9d8W3vHfytdREZeVmAfxToKFRnO5zxX9g/m6N5hiEBceMc8