How this calculator works
Circle theorems describe fixed relationships between angles and lengths around a circle. The inscribed angle theorem says an angle at the circumference is half the central angle standing on the same arc, so the central angle is twice the inscribed angle. Thales theorem is the special case where the arc is a semicircle: the angle at the circumference is a right angle, and the triangle formed with the diameter as hypotenuse obeys Pythagoras. In a cyclic quadrilateral, all four vertices lie on one circle and opposite angles add to 180 degrees. For two secants drawn from the same external point, the product of the near and whole lengths is the same for both lines. For a tangent and a secant from one point, the tangent length squared equals the near part times the whole secant. The calculator applies whichever relationship you choose and reports the remaining values.
Formula
Inscribed angle: central angle = 2θ. Thales: the angle is 90 degrees and hypotenuse² = leg² + other leg². Cyclic quadrilateral: A + C = 180 degrees and B + D = 180 degrees. Intersecting secants: PA × PB = PC × PD. Tangent-secant: tangent² = external part × whole secant.
Worked example
An inscribed angle of 40 degrees stands on a central angle of 80 degrees. With a radius of 5 m, its arc is 80 degrees in radians times 5, about 6.981 m, and the sector area is about 17.453 m². A secant with a near part of 4 m and a whole length of 10 m, paired with a near part of 5 m, gives a whole second secant of 8 m and an inside part of 3 m. A 6 m tangent with a 4 m external secant part gives a whole secant of 9 m.
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Frequently asked questions
Why is the central angle twice the inscribed angle?
Both angles stand on the same arc. The central angle measures the arc directly in radians, while the inscribed angle is measured at a point on the circumference. Drawing the diameter through the inscribed vertex splits the figure into two isosceles triangles whose angles show the factor of two.
What is special about an angle in a semicircle?
The diameter subtends a half turn at the centre, so the inscribed angle is half of that, or 90 degrees. That is why a triangle built on a diameter with its third vertex anywhere on the circle is always a right triangle.
Do the secant products use the whole secant or only the inside part?
Use the near part from the external point to the first intersection, and the whole length from that point through to the far intersection. The product is near times whole. Multiplying the two inside parts would give a different theorem and is not what this page calculates.
Why is the tangent theorem a limiting case of the secant theorem?
Imagine the second secant moving until its two intersections merge at the tangent point. Both near and whole lengths become the tangent length, so their product becomes the square of the tangent. The same power-of-a-point relationship therefore covers both cases.