SAT® · Math: Geometry and Trigonometry
The Distance Formula in Circle Problems
The distance formula measures the gap between two points using the Pythagorean theorem on their x and y differences. On circle problems, it tells you how far a point sits from the center, which decides the circle's equation and whether that point is inside, on, or outside it.
Plot two points and draw a right triangle: one leg is the horizontal gap between them, the other leg is the vertical gap. The straight-line distance between the points is the hypotenuse. That's the whole formula, distance = √((x2-x1)2 + (y2-y1)2), it's just a2 + b2 = c2 with coordinates for labels. Nothing new to memorize once you see the triangle hiding in the grid.
That same idea decides where a point sits relative to a circle. A circle is just every point sitting exactly one fixed distance, the radius, from a center. So take any point, find its distance to the center using the formula, and compare that number to the radius. Distance greater than radius means outside, equal means right on the circle, less means inside. Squaring both sides of the distance formula and setting it equal to r2 is exactly how the standard circle equation (x-h)2 + (y-k)2 = r2 gets built.
This connects straight to writing circle equations from a diameter's endpoints: the center is the midpoint of the diameter, and the radius is the distance from that midpoint to either endpoint. Once you have both, the equation falls out directly. The same triangle-on-a-grid thinking also backs up slope and midpoint questions elsewhere in this domain, so getting comfortable with legs-and-hypotenuse here pays off across the whole coordinate geometry stretch of the test.
Key terms
- Distance formula
- The formula √((x2-x1)2+(y2-y1)2), giving the straight-line gap between two points.
- Midpoint
- The point exactly halfway between two given points, found by averaging their x-values and their y-values.
- Radius
- The fixed distance from a circle's center to any point on the circle.
- Standard circle equation
- The equation (x-h)2+(y-k)2=r2, where (h,k) is the center and r is the radius.
Watch it explained
- How it worksHow does the distance from a point to a center decide whether that point lies inside or outside?
- Worked exampleFind the distance between the points (2, -3) and (8, 5).
- Worked exampleFind the equation of the circle whose diameter runs from (1, 2) to (7, 10).
- Common mistakeWhy isn't the distance formula a brand new rule you have to memorize on its own?
- Ties togetherHow does one right triangle drawn on the grid link the circle equation to Pythagoras?
Check yourself
End of topic.