Circle Calculator: Area, Sector, Segment, Annulus, Ellipse and the Area Between Two Semicircles
Find the area and lengths of circles, sectors, segments, rings and ellipses, and solve the semicircle puzzle.
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How this calculator works
Choose the shape and enter the radius, a second radius if there is one, and an angle for a sector or segment.
The calculator gives the area, the arc, the chord and the perimeter as they apply.
For the semicircle puzzle enter the large radius 4 and the small radius 2. The green S-shape is two curved bands, each half the area of a ring, so the total is pi (16 - 4) = 12 pi, about 37.70.
Formulas used
- A = \pi r^2,\quad C = 2\pi r
- A_{sector} = \tfrac12 r^2\theta
- A_{segment} = \tfrac12 r^2(\theta - \sin\theta)
- A_{ring} = \pi(R^2 - r^2)
- A_{S} = \pi(R^2 - r^2)
Worked example
With the default numbers in the calculator above:
- Area of the large semicircle: 25.13
- Area of the small semicircle cut out of it: 6.28
- Area of one curved band: 18.85
- Area of both bands, which is the S-shaped (green) region: 37.70 (12.00 pi)
Change the numbers in the calculator to see the full working, every step, for your own case.
A second worked example
A different case, with these numbers: What do you want? Circle.
- Area: 50.27 square units
- Circumference: 25.13
- Diameter: 8.00
Common questions
What is the answer to the semicircle puzzle?
With AP = PQ = QB = 4 cm the big semicircles have radius 4 and the small ones radius 2. Each green band is half pi (16 - 4) = 6 pi, so the green region is 12 pi, about 37.7 square cm.
Does the big outer circle matter?
No. Only the four semicircles make the green region, so the outer circle is not used.
How is a segment different from a sector?
A sector is a slice from the centre. A segment is the piece cut off by a chord, which is the sector less the triangle.
Is the ellipse perimeter exact?
No. There is no simple exact formula. Ramanujan's approximation is within a few parts per million for any ellipse.
Related calculators
Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any school or university mathematics text (areas of circles, sectors and segments).
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