Interior vs Exterior Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
A point P=(3,2)P = (3, 2) and a circle centered at (1,1)(1, 1) with radius r=3r = 3. Is PP interior or exterior to the circle? Show your work using the distance formula.

Solution

  1. 1
    Step 1: Use the distance formula to find the distance from P=(3,2)P = (3,2) to the center (1,1)(1,1): d=(3−1)2+(2−1)2=4+1=5d = \sqrt{(3-1)^2 + (2-1)^2} = \sqrt{4 + 1} = \sqrt{5}.
  2. 2
    Step 2: Compare dd to the radius r=3r = 3. We have 5≈2.236<3\sqrt{5} \approx 2.236 < 3.
  3. 3
    Step 3: Since d<rd < r, the point lies inside the circle.

Answer

PP is interior to the circle (since 5<3\sqrt{5} < 3).
To determine if a point is inside or outside a circle, compute the distance from the point to the center and compare it to the radius. If the distance is less than the radius, the point is interior; if greater, exterior; if equal, it's on the boundary.

About Interior vs Exterior

Interior consists of points strictly inside a boundary; exterior consists of points strictly outside the boundary.

Learn more about Interior vs Exterior →

More Interior vs Exterior Examples

Example 1 easy

A circle is drawn on a piece of paper. Point [formula] is 3 cm from the center and the radius is 5 c

Example 2 medium

A rectangle has vertices at [formula], [formula], [formula], and [formula]. Determine whether the po

Example 3 easy

A triangle is drawn on paper. Point [formula] is inside the triangle and point [formula] is outside.

← All Interior vs Exterior Examples Interior vs Exterior Concept

Related Concepts

BoundaryArea