Standard Equation of a Circle Calculator
Enter a circle's centre (h, k) and radius r to get its standard equation (x−h)²+(y−k)²=r², area, circumference, and x- and y-intercepts instantly.
- 1
Radius squared r²
5 × 5 = 25 - 2
Area = π × r²
π × 25 = 78.5398 - 3
Circumference = 2π × r
2 × π × 5 = 31.4159
How does this calculator work?
For a circle with centre (h, k) and radius r, the standard equation is (x−h)²+(y−k)²=r². Area = πr², circumference = 2πr. x-intercepts exist when r² > k², at x = h ± √(r²−k²); y-intercepts when r² > h², at y = k ± √(r²−h²). Example: centre (3,−2), r=5 → (x−3)²+(y+2)²=25.
Formula
How this is calculated
The standard (centre-radius) form of a circle, (x − h)² + (y − k)² = r², says that every point (x, y) on the circle is exactly r units from the centre (h, k) — that distance is just the Pythagorean theorem applied to the horizontal and vertical separations. It is the most readable form because h, k and r appear directly, with no expansion required.
The x-intercepts are found by substituting y = 0: (x − h)² = r² − k². If r² > k², two intercepts exist at x = h ± √(r² − k²); if r² = k² the circle just touches the x-axis at a single tangent point; and if r² < k² the circle sits entirely above or below the x-axis with no real intercepts. The y-intercepts follow symmetrically by substituting x = 0.
This form is equivalent to the general form x² + y² + Dx + Ey + F = 0, where D = −2h, E = −2k and F = h² + k² − r². You can convert between them by expanding the squares (standard → general) or completing the square (general → standard).
Frequently asked questions
(x − h)² + (y − k)² = r², where (h, k) is the centre and r is the radius. Every point on the circle is exactly r units from the centre, which follows from the Pythagorean theorem.
Set y = 0: (x − h)² = r² − k². If r² − k² is positive, the intercepts are x = h ± √(r² − k²). If it equals zero, the circle is tangent to the x-axis at x = h. If it is negative, there are no x-intercepts.
Standard form (x−h)²+(y−k)²=r² shows centre and radius directly. General form x²+y²+Dx+Ey+F=0 collects everything on the left; the centre is (−D/2, −E/2) and radius = √(D²/4 + E²/4 − F). Standard form is easier to read; general form is useful in linear-algebra or curve-fitting contexts.
Also known as
TG we-Calculate Editorial Team. (2026). Standard Equation of a Circle Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/standard-equation-circle-calculator
TG we-Calculate Editorial Team. "Standard Equation of a Circle Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/standard-equation-circle-calculator.
TG we-Calculate Editorial Team, "Standard Equation of a Circle Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/standard-equation-circle-calculator
@misc{wecalculate_standard_equation_circle_calculator, title = {Standard Equation of a Circle Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/standard-equation-circle-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?