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 boundary is the exact dividing line between the inside of a region and the outside.
Common stuck point:The procedure for boundary is the easy part; the trap is treating the interior as part of the boundary. Asking "Am I asking about the dividing line itself, not the space it encloses?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Am I asking about the dividing line itself, not the space it encloses?
Worked Examples
Example 1
easy
Draw a circle. What is the boundary of the circle (the disk region)? What is the interior? Use everyday language.
Answer
Boundary = the circle (curved edge); interior = all points strictly inside.
First step
1
Step 1: The disk is the filled-in circle — all points inside and on the edge.
Full solution
2
Step 2: The boundary is the curved line (the circle itself) — the edge that separates the inside from the outside.
3
Step 3: The interior is everything strictly inside the boundary — all points whose distance from the centre is less than the radius.
4
Step 4: Points outside the circle are the exterior — not part of the disk.
The boundary of a region is the set of points that separate the inside from the outside. For a disk, the boundary is the circle. Touching the boundary means you are on the edge — neither fully inside nor outside.
Example 2
medium
What is the boundary of a square with vertices (0,0), (4,0), (4,4), (0,4)? Find the perimeter of the boundary and determine whether the point (4,2) is on the boundary, interior, or exterior.The boundary of the square is its four sides; total perimeter = 16
Example 3
medium
A 10 m by 6 m rectangle has a 2 m square cut out of one corner. Find the boundary length of the resulting L-shape.
Example 4
hard
A regular hexagon has boundary length 30. Find one side.
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
A country is enclosed by a border. Name what the border represents geometrically (boundary, interior, or exterior) for each region: (a) inside the country, (b) the border itself, (c) the neighbouring country.
Example 2
medium
The region R is defined by the inequalities x≥0, y≥0, and x+y≤6. Describe the boundary of R and find its perimeter.Boundary of the region R: triangle with vertices (0,0), (6,0), (0,6); perimeter
Example 3
easy
What is the boundary of a circle (the disk)?
Example 4
easy
A fence around a yard represents what part of the region?
Example 5
easy
What is the boundary of a square region?
Example 6
easy
Is a point exactly on the edge of a region inside, outside, or on the boundary?
Example 7
easy
The length of a region's boundary is called what for a polygon?
Example 8
easy
Does the boundary of a region have area?
Example 9
easy
What is the boundary of a line segment?
Example 10
easy
Which has a longer boundary: a circle of radius 5 or a circle of radius 2?
Example 11
medium
Why is the boundary of a region one dimension lower than the region itself?
Example 12
medium
What is the boundary of a solid cube (a 3D region)?
Example 13
medium
A square garden of side 10 m has a path built along its boundary. How long is the path?Boundary (perimeter) of the square garden
Example 14
medium
Two regions share part of their boundary (a common wall). What is that shared boundary called?
Example 15
medium
A closed curve divides the plane into how many regions (besides the curve itself)?
Example 16
medium
A rectangular field is 20 m by 15 m. Fencing costs $4 per meter. What does fencing the boundary cost?Boundary (perimeter) = 2(20 + 15) = 70 m; fencing cost
Example 17
medium
Does a region always have to be enclosed by a single boundary curve?
Example 18
medium
A circular garden has radius 7 m. What length of edging is needed for its boundary (use π≈3.14)?
Example 19
challenge
An L-shaped region is made by removing a 2×2 square corner from a 6×6 square. Find the perimeter (boundary length).
Example 20
challenge
Why can two regions have the same area but very different boundary lengths?
Example 21
challenge
A coastline appears longer the more closely you measure it. What does this reveal about some boundaries?
Example 22
challenge
Explain why the boundary of a region is itself a closed curve (has no boundary of its own), for a simple 2D region.
Example 23
easy
A triangular flag has sides 5, 7, and 8 inches. How long is its boundary?
Example 24
easy
A circular pool has diameter 10 m. What is the length of its boundary (use π≈3.14)?
Example 25
easy
A pentagonal park has sides 10, 12, 15, 11, and 14 m. Find its boundary length.
Example 26
medium
A square garden has side 8 m. A path of width 1 m runs along the inside boundary. What is the inner edge's length?
Example 27
medium
A ring (annulus) has outer radius 10 cm and inner radius 4 cm. Find the total boundary length (use π≈3.14).
Example 28
medium
A 5 cm by 8 cm rectangle has fencing along its boundary at $3 per cm. What is the cost?
Example 29
medium
Two rectangles share one common 4 m edge. Rectangle A is 4 by 6 m, rectangle B is 4 by 5 m. What is the perimeter of the combined region?
Example 30
medium
A semicircular region has diameter 14 cm. Find the total boundary length (use π≈3.14).
Example 31
medium
A 12 m by 8 m yard has a square 3-by-3 m flowerbed inside it. What is the total boundary length, including the flowerbed?
Example 32
medium
A right triangle has legs 6 and 8. Find its boundary length.
Example 33
hard
Among all rectangles with area 36, which has the smallest boundary length?
Example 34
hard
A T-shape has top bar 8 wide, 2 tall and a vertical stem 3 wide, 6 tall centered below. Find its boundary length.
Example 35
hard
An equilateral triangle has perimeter (boundary length) 30 cm. Find one side and the height.
Example 36
hard
A 200 m running track is made of two straight sides of length s and two semicircular ends of radius 25 m. Find s (use π≈3.14).
Example 37
hard
A 30 m by 20 m rectangular field is enclosed by fence. A diagonal fence is added. What is the total fence length used (use 1300≈36.06)?
Example 38
challenge
A shape is built from a unit square plus a unit square attached on its top edge, plus a unit square attached to the right of the top one. Find the boundary length.
Example 39
challenge
A rope of length 40 m is shaped into a closed curve enclosing maximum area. What is the shape and the area (use π≈3.14)?