Interior vs Exterior Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Interior vs Exterior.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

A closed fence divides the world into two zones: the yard inside and everything else outside. Any closed curve does the same—splitting the plane into an interior region and an exterior region.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A closed curve divides the plane into exactly two regions: interior and exterior.

Common stuck point: Points on the boundary belong to neither the interior nor the exterior—they form a separate category.

Sense of Study hint: When determining if a point is interior or exterior, draw a ray from the point outward. Count how many times it crosses the boundary: an odd number means interior, an even number means exterior.

Worked Examples

Example 1

easy
A circle is drawn on a piece of paper. Point A is 3 cm from the center and the radius is 5 cm. Point B is 7 cm from the center. Which point is interior and which is exterior?

Solution

  1. 1
    Step 1: Identify the boundary. The circle has radius r = 5 cm, so the boundary is the set of all points exactly 5 cm from the center.
  2. 2
    Step 2: Check Point A. Since 3 < 5, Point A is closer to the center than the radius, so it lies inside the circle — it is an interior point.
  3. 3
    Step 3: Check Point B. Since 7 > 5, Point B is farther from the center than the radius, so it lies outside the circle — it is an exterior point.

Answer

Point A is interior; Point B is exterior.
A point is interior to a region if it lies strictly inside the boundary, and exterior if it lies strictly outside. For a circle, compare each point's distance from the center to the radius: less than r means interior, greater than r means exterior, equal to r means on the boundary.

Example 2

medium
A rectangle has vertices at (0,0), (4,0), (4,3), and (0,3). Determine whether the point (2, 1.5) is interior, exterior, or on the boundary of the rectangle.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A triangle is drawn on paper. Point P is inside the triangle and point Q is outside. If you draw a straight line from P to Q, how many times must it cross the boundary of the triangle?

Example 2

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

BoundaryArea

Background Knowledge

These ideas may be useful before you work through the harder examples.

boundary