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Circle (x − h)² + (y − k)² = r²

Conic Sections · Class XI

Move the center (h, k) and resize the radius r. Both halves of the circle are plotted.

0
type a value
-55
0
type a value
-55
3
type a value
0.16
Live values
  • Circumference18.8496
  • Area28.2743
  • Distance from O0
xy
  • Upper half
  • Lower half

Formulas in this lab

  • Circumference
    2πr2\pi r
  • Area
    πr2\pi r^2
  • Distance from O
    h2+k2\sqrt{h^2+k^2}
Tip: A circle is an ellipse with a = b , see the ellipse lab.

Frequently asked questions

▶What is the standard equation of a circle?

A circle with centre $(h, k)$ and radius $r$ has equation $(x - h)^2 + (y - k)^2 = r^2$. Expanded form is $x^2 + y^2 + Dx + Ey + F = 0$ with centre $(-D/2, -E/2)$ and radius $\sqrt{D^2/4 + E^2/4 - F}$.

▶How do I use the circle equation lab?

Slide centre coordinates $(h, k)$ and radius $r$. The lab plots the circle and reports area $\pi r^2$ and circumference $2\pi r$. Try $(0, 0)$ and $r = 5$ to see the classic $x^2 + y^2 = 25$.

▶How do I recognize a circle from a general second-degree equation?

If the coefficients of $x^2$ and $y^2$ are equal and there is no $xy$ term, you have a circle. Complete the square to find the centre and radius. JEE asks: is $2x^2 + 2y^2 - 4x + 6y - 1 = 0$ a circle? Divide by $2$ first; then yes.

▶Where do circles appear in real life?

Wheel design (every bike and Maruti car wheel), pizza serving sizes, archery targets, and the orbits of geostationary satellites at $42{,}164$ km radius from Earth's centre. The equation lets engineers compute precise radii from physical constraints.