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 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:Interior is the space strictly inside a boundary, exterior is everything strictly outside it.
Common stuck point:The procedure for interior vs exterior is the easy part; the trap is forgetting the curve must be closed. Asking "Am I deciding whether a point is strictly inside or strictly outside a closed boundary?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Am I deciding whether a point is strictly inside or strictly outside a closed boundary?
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?
Answer
Point A is interior; Point B is exterior.
First step
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.
Full solution
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
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.
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.Rectangle with vertices (0,0), (4,0), (4,3), (0,3) — is (2, 1.5) inside?
Example 3
easy
A triangle has vertices at (0,0),(6,0),(0,4). Determine whether (1,1) is inside.Triangle with vertices (0,0), (6,0), (0,4) — is (1,1) interior?
Example 4
medium
A rectangle is bounded by 0≤x≤8 and 0≤y≤5. Classify (8,3): interior, on, or exterior.Rectangle 0≤x≤8, 0≤y≤5 — classify (8, 3): interior, boundary, or exterior?
Example 5
medium
The unit circle is x2+y2=1. Sketch and shade the EXTERIOR region. Write the inequality.
Example 6
hard
Using the ray-casting test: a polygon is drawn in the plane. From a candidate point, shoot a ray to the right. The ray crosses the polygon boundary 4 times. Interior or exterior?
Example 7
hard
A circle is described by the inequality (x−2)2+(y−3)2≤16. Describe the interior as a strict inequality.
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.
Example 3
easy
A point is 2 units from the center of a circle of radius 5. Is it interior or exterior?
Example 4
easy
A point is 8 units from the center of a circle of radius 5. Interior or exterior?
Example 5
easy
A point is exactly 5 units from the center of a circle of radius 5. Where is it?
Example 6
easy
A closed fence divides the world into how many zones?
Example 7
easy
Does a boundary point belong to the interior?
Example 8
easy
Inside a square room, are you in the interior or exterior of the square?
Example 9
easy
For a circle of radius r, give the condition for a point at distance d to be exterior.
Example 10
easy
Does the inside of a region have area?
Example 11
medium
Is the point (3,4) interior, on, or exterior to the circle x2+y2=25?
Example 12
medium
Is (1,1) interior or exterior to the circle x2+y2=25?
Example 13
medium
Why does a closed curve always separate the plane into a distinct inside and outside?
Example 14
medium
To test if a point is inside a complicated polygon, you can draw a ray outward and count crossings. An odd number of crossings means what?
Example 15
medium
A point is inside a square room but on the doorway threshold line. Is it interior, exterior, or boundary?
Example 16
medium
A circle of radius 3 sits inside a circle of radius 7 (same center). Where is a point 5 units from the center — between the circles, inside both, or outside both?
Example 17
medium
Why is the interior of a circle called an 'open' region, while including the boundary makes it 'closed'?
Example 18
medium
Is the center of a circle interior or exterior?
Example 19
challenge
Is the point (6,8) interior, on, or exterior to the circle (x−2)2+(y−3)2=36?
Example 20
challenge
A maze is drawn as a single wiggly closed curve. A treasure dot is somewhere in the picture. How can you tell if it's inside the curve without tracing the whole maze?
Example 21
challenge
A region has a hole in it (like a donut). Is a point in the hole interior or exterior to the region?
Example 22
challenge
Why does the boundary belong to neither the interior nor the exterior, making three distinct categories?
Example 23
easy
A square has side length 6. A point is 4 units from the center. Could it be interior, exterior, or on the boundary?
Example 24
easy
For the circle x2+y2=16, is the origin interior, on, or exterior?
Example 25
easy
Is (2,1) interior, on, or exterior to the circle x2+y2=9?
Example 26
easy
Is (0,5) interior, on, or exterior to the circle x2+y2=25?
Example 27
medium
For the circle (x−3)2+(y+2)2=25, is (7,1) interior, on, or exterior?
Example 28
medium
For the circle (x+1)2+(y−2)2=49, is (2,5) interior?
Example 29
medium
Two concentric circles have radii 3 and 7. Where is a point 5 units from the center?
Example 30
medium
A point P lies on the perimeter of a square. Is P interior, exterior, or boundary?Point P lies on the perimeter — boundary, not interior or exterior
Example 31
medium
To test if (4,−1) is inside the polygon with vertices (0,0),(5,0),(5,5),(0,5), what do you check?Square polygon (0,0)–(5,5) — point (4, −1) is below the bottom edge, so it is exterior
Example 32
medium
Is the point (−2,−3) inside the circle of radius 4 centered at the origin?
Example 33
hard
For the ellipse 9x2+4y2=1, is (2,1) interior?
Example 34
hard
A triangle has vertices (0,0),(4,0),(2,3). Is (2,1) interior?Triangle with vertices (0,0), (4,0), (2,3) — is (2,1) an interior point?
Example 35
hard
Is the point (1,1) in the interior of the region defined by x+y<3 AND x2+y2<5?
Example 36
hard
A square room of side 10 has a smaller square pillar of side 2 in the exact center. A point sits 4 units from the room's center. Is it interior to the walkable region?
Example 37
hard
Is the point (0,0) in the interior, on, or exterior to the parabola region y≥x2?
Example 38
challenge
Two squares overlap to form an L-shape. A point lies inside one square but not the other. Is it interior, exterior, or boundary of the L-shape?L-shape from two squares — a point inside only one square is still interior to the L-shaped region
Example 39
challenge
A donut shape (annulus) has inner radius 2 and outer radius 5. Where is the point (3,4)? (Centered at origin.)