Cut a Corner Out of a Plot and the Fence Stays the Same Length
Learn to find the perimeter of polygons and composite shapes, the circumference of a circle both ways, the length of an arc and the perimeter of a sector, and to solve running-track and fencing problems.
Does cutting a piece out of a plot make the fence shorter?
A rectangular plot is m by m, so its fence measures
Now cut a m by m square out of one corner. The plot has lost of land. How much fence does it save?
None at all. Walk the new boundary: m along the bottom, m up the right side, m in, m up, m along the top, m down the left side.
The same m. The two new edges going inward are exactly as long as the two edges they replaced — the m step in matches the m removed from the top, and the m step up matches the m removed from the side.
So area and perimeter are independent. Losing area does not have to cost boundary, and gaining boundary does not have to gain area. That is worth having straight before any composite-shape question, because the instinct that a smaller shape needs a shorter fence is wrong often enough to be dangerous.
This page covers the first part of the CBSE Class 9 Mathematics chapter on measuring space — perimeters of polygons, circumference, arc length, sector perimeter, and the composite problems that combine straight edges with curves.
Now cut a m by m square out of one corner. The plot has lost of land. How much fence does it save?
None at all. Walk the new boundary: m along the bottom, m up the right side, m in, m up, m along the top, m down the left side.
The same m. The two new edges going inward are exactly as long as the two edges they replaced — the m step in matches the m removed from the top, and the m step up matches the m removed from the side.
So area and perimeter are independent. Losing area does not have to cost boundary, and gaining boundary does not have to gain area. That is worth having straight before any composite-shape question, because the instinct that a smaller shape needs a shorter fence is wrong often enough to be dangerous.
This page covers the first part of the CBSE Class 9 Mathematics chapter on measuring space — perimeters of polygons, circumference, arc length, sector perimeter, and the composite problems that combine straight edges with curves.
How do you find the perimeter of a polygon or a composite shape?
Add up every edge of the boundary, once each. For a regular polygon that collapses to one multiplication.
For a regular polygon of sides each of length :
Worked example 1. A regular hexagon of side cm has perimeter cm. A regular octagon of side cm has cm.
Worked example 2 — an irregular polygon. A five-sided plot with edges m, m, m, m and m has perimeter
Worked example 3 — a missing side. A quadrilateral has perimeter cm with three sides of cm, cm and cm. The fourth is
Worked example 4 — an L-shaped room. A room is m by m with a m by m alcove removed from a corner. Walking the boundary: , then , then , then , then , then .
Check against the uncut rectangle: m. The same, by the argument this page opened with.
Worked example 5 — where the shortcut fails. Now cut a m by m notch out of the middle of one long wall rather than a corner. The boundary gains three new edges — m in, m across, m out — while losing only the m of wall they span:
So the corner case and the middle case behave differently, and the rule cut-outs do not change the perimeter holds only at a corner, where the two new edges replace two equal ones.
The reliable method is always to walk the boundary and list the edges. It takes a few seconds longer than reaching for and it works on every shape, including the ones where the shortcut quietly gives the wrong answer.
One habit that prevents double counting. Mark each edge with a tick as you add it. An internal line in a figure is not part of the boundary — a diagonal drawn to help find an area contributes nothing to the perimeter, and including it is the most common error in composite questions.
For a regular polygon of sides each of length :
Worked example 1. A regular hexagon of side cm has perimeter cm. A regular octagon of side cm has cm.
Worked example 2 — an irregular polygon. A five-sided plot with edges m, m, m, m and m has perimeter
Worked example 3 — a missing side. A quadrilateral has perimeter cm with three sides of cm, cm and cm. The fourth is
Worked example 4 — an L-shaped room. A room is m by m with a m by m alcove removed from a corner. Walking the boundary: , then , then , then , then , then .
Check against the uncut rectangle: m. The same, by the argument this page opened with.
Worked example 5 — where the shortcut fails. Now cut a m by m notch out of the middle of one long wall rather than a corner. The boundary gains three new edges — m in, m across, m out — while losing only the m of wall they span:
So the corner case and the middle case behave differently, and the rule cut-outs do not change the perimeter holds only at a corner, where the two new edges replace two equal ones.
The reliable method is always to walk the boundary and list the edges. It takes a few seconds longer than reaching for and it works on every shape, including the ones where the shortcut quietly gives the wrong answer.
One habit that prevents double counting. Mark each edge with a tick as you add it. An internal line in a figure is not part of the boundary — a diagonal drawn to help find an area contributes nothing to the perimeter, and including it is the most common error in composite questions.
Formula
How do you find a circle's circumference, and its radius from the circumference?
**The circumference is times the diameter:**
and rearranging gives the radius from a known circumference:
Take whenever the radius is a multiple of , and otherwise.
Worked example 1 — circumference from the radius. A circle of radius cm:
Worked example 2 — from the diameter. A circle of diameter cm has cm:
Worked example 3 — radius from the circumference. A circular track measures m around:
Check: m. Correct.
Worked example 4 — diameter from the circumference. A tree's trunk measures cm around, so
Worked example 5 — a wheel and its revolutions. A wheel of radius cm has
One revolution covers m, so revolutions cover
**Why is the same for every circle.** Measure the circumference and the diameter of any round object and divide. A bangle, a bucket and a cartwheel all give about — the ratio is a property of circles as a shape, not of any particular circle. That constancy is what makes a single formula possible.
** is an approximation, and the answer should say so.** Write *taking * at the start. is irrational, as the second part of the numbers chapter established, so no fraction is exactly right and merely agrees to two decimal places.
**The perimeter of a semicircular region is not .** The curved part is half the circumference, , but the boundary also runs back along the diameter:
For cm that is cm. Forgetting the straight edge is the standard error, and it happens because the formula for the curve is the part that had to be looked up.
and rearranging gives the radius from a known circumference:
Take whenever the radius is a multiple of , and otherwise.
Worked example 1 — circumference from the radius. A circle of radius cm:
Worked example 2 — from the diameter. A circle of diameter cm has cm:
Worked example 3 — radius from the circumference. A circular track measures m around:
Check: m. Correct.
Worked example 4 — diameter from the circumference. A tree's trunk measures cm around, so
Worked example 5 — a wheel and its revolutions. A wheel of radius cm has
One revolution covers m, so revolutions cover
**Why is the same for every circle.** Measure the circumference and the diameter of any round object and divide. A bangle, a bucket and a cartwheel all give about — the ratio is a property of circles as a shape, not of any particular circle. That constancy is what makes a single formula possible.
** is an approximation, and the answer should say so.** Write *taking * at the start. is irrational, as the second part of the numbers chapter established, so no fraction is exactly right and merely agrees to two decimal places.
**The perimeter of a semicircular region is not .** The curved part is half the circumference, , but the boundary also runs back along the diameter:
For cm that is cm. Forgetting the straight edge is the standard error, and it happens because the formula for the curve is the part that had to be looked up.
How long is an arc that subtends a given angle?
It is that fraction of the whole circumference, with the fraction given by the angle out of :
And the perimeter of a sector adds the two radii that bound it:
Worked example 1 — a sixth of a circle. In a circle of radius cm, an arc subtends . The full circumference is cm, so
The sector's perimeter adds two radii:
Worked example 2 — a quadrant. In a circle of radius cm, the circumference is cm. A arc is a quarter:
Worked example 3 — a semicircle as a sector. With the formula gives arc and perimeter , matching the previous section. The sector formula contains the semicircle as a special case, so there is only one thing to remember.
Worked example 4 — finding the angle from the arc. An arc of cm lies on a circle of radius cm, whose circumference is cm. Then
Worked example 5 — finding the radius from the arc and angle. A arc measures cm. Since is one third of a turn, the full circumference is cm, so
Worked example 6 — a fan-shaped park. A sector-shaped lawn of radius m subtends . The circumference would be m, so
Fencing the lawn needs m.
The perimeter of a sector and the length of its arc are different questions. The arc is m; the fence is m. A question asking for the fencing wants the radii included, and one asking for the arc does not. Reading which is wanted costs nothing and mixing them up costs the whole answer.
And the perimeter of a sector adds the two radii that bound it:
Worked example 1 — a sixth of a circle. In a circle of radius cm, an arc subtends . The full circumference is cm, so
The sector's perimeter adds two radii:
Worked example 2 — a quadrant. In a circle of radius cm, the circumference is cm. A arc is a quarter:
Worked example 3 — a semicircle as a sector. With the formula gives arc and perimeter , matching the previous section. The sector formula contains the semicircle as a special case, so there is only one thing to remember.
Worked example 4 — finding the angle from the arc. An arc of cm lies on a circle of radius cm, whose circumference is cm. Then
Worked example 5 — finding the radius from the arc and angle. A arc measures cm. Since is one third of a turn, the full circumference is cm, so
Worked example 6 — a fan-shaped park. A sector-shaped lawn of radius m subtends . The circumference would be m, so
Fencing the lawn needs m.
The perimeter of a sector and the length of its arc are different questions. The arc is m; the fence is m. A question asking for the fencing wants the radii included, and one asking for the arc does not. Reading which is wanted costs nothing and mixing them up costs the whole answer.
How do you find the perimeter of a running track?
Split the boundary into straight pieces and curved pieces, work out each, and add. The curves usually combine into a whole number of circles.
Worked example 1 — a classic track. A running track has two straights of m joined by two semicircular ends, each of diameter m.
The two semicircles together make one full circle of diameter m:
Adding the straights:
The two semicircular ends are the same size, so they add to one circle — noticing that saves halving anything.
Worked example 2 — the cost of fencing it. At per metre,
Worked example 3 — a semicircular flower bed on a rectangle. A plot is m by m with a semicircular bed attached to one short side, using that side as its diameter. So m and m. The boundary is three sides of the rectangle plus the semicircular arc — the short side it sits on is no longer a boundary:
rounded to two decimal places.
Worked example 4 — a quadrant cut from a square corner. A square of side cm has a quadrant of radius cm removed, centred at one corner. The boundary is two full sides plus the quadrant arc:
Worked example 5 — a shape made of three semicircles. A large semicircle of diameter cm sits above two smaller semicircles of diameter cm each, drawn on its diameter.
- large arc: cm
- each small arc: cm, so cm for the two
Check the striking part: the two small arcs together are the same length as the one large arc. Halving a diameter halves its semicircular arc, and there are two of them — so the total is unchanged.
The edge that disappears is what these problems test. When a semicircle is attached along a side, that side stops being boundary. When a quadrant is cut out of a corner, two part-sides stop being boundary and an arc takes their place. Walk the boundary with a finger and only add what you actually travel along, which is the same discipline the composite-polygon section needed and the reason to build it early.
Worked example 1 — a classic track. A running track has two straights of m joined by two semicircular ends, each of diameter m.
The two semicircles together make one full circle of diameter m:
Adding the straights:
The two semicircular ends are the same size, so they add to one circle — noticing that saves halving anything.
Worked example 2 — the cost of fencing it. At per metre,
Worked example 3 — a semicircular flower bed on a rectangle. A plot is m by m with a semicircular bed attached to one short side, using that side as its diameter. So m and m. The boundary is three sides of the rectangle plus the semicircular arc — the short side it sits on is no longer a boundary:
rounded to two decimal places.
Worked example 4 — a quadrant cut from a square corner. A square of side cm has a quadrant of radius cm removed, centred at one corner. The boundary is two full sides plus the quadrant arc:
Worked example 5 — a shape made of three semicircles. A large semicircle of diameter cm sits above two smaller semicircles of diameter cm each, drawn on its diameter.
- large arc: cm
- each small arc: cm, so cm for the two
Check the striking part: the two small arcs together are the same length as the one large arc. Halving a diameter halves its semicircular arc, and there are two of them — so the total is unchanged.
The edge that disappears is what these problems test. When a semicircle is attached along a side, that side stops being boundary. When a quadrant is cut out of a corner, two part-sides stop being boundary and an arc takes their place. Walk the boundary with a finger and only add what you actually travel along, which is the same discipline the composite-polygon section needed and the reason to build it early.
Exam tip
Exam tip: walk the boundary, and say which value of pi you used
Trace the boundary with a finger and list each edge before adding anything. A side that a semicircle is attached to, or that a cut-out replaced, is not boundary any more.
Never include an internal line. A diagonal drawn to help with area contributes nothing to the perimeter.
**State your **: write *taking * at the start. Use when the radius is a multiple of , and otherwise.
****, and backwards : a track of m gives m. Always check by substituting back.
**A semicircular region's perimeter is ** — the diameter is part of the boundary. For : cm.
**Arc length is , and the sector perimeter adds **. Radius with gives an arc of cm and a perimeter of cm.
Read whether the question wants the arc or the whole sector boundary — m against m in the park example.
Two equal semicircular ends make one full circle — use directly instead of halving twice.
Cutting a corner out does not change the perimeter; cutting a notch from the middle of a side does. Do not assume either — walk the boundary.
Keep the units consistent and convert before multiplying: a wheel of cm is m, so turns cover m.
And for cost problems, perimeter times rate: .
Never include an internal line. A diagonal drawn to help with area contributes nothing to the perimeter.
**State your **: write *taking * at the start. Use when the radius is a multiple of , and otherwise.
****, and backwards : a track of m gives m. Always check by substituting back.
**A semicircular region's perimeter is ** — the diameter is part of the boundary. For : cm.
**Arc length is , and the sector perimeter adds **. Radius with gives an arc of cm and a perimeter of cm.
Read whether the question wants the arc or the whole sector boundary — m against m in the park example.
Two equal semicircular ends make one full circle — use directly instead of halving twice.
Cutting a corner out does not change the perimeter; cutting a notch from the middle of a side does. Do not assume either — walk the boundary.
Keep the units consistent and convert before multiplying: a wheel of cm is m, so turns cover m.
And for cost problems, perimeter times rate: .
Did you know
Why a fixed length of fence can enclose almost nothing
The plot on this page kept its m fence after losing of land. Push that idea further and it becomes startling.
Take the m by m rectangle and cut a long thin notch into it from one side — say m deep and m wide. The area lost is only , but the boundary gains m and loses m, so the fence grows by m. **Almost no land lost, m of extra fence needed.**
Repeat with ten such notches and the fence passes m while the plot is still very nearly . There is no limit: a region of fixed area can be given as long a boundary as you like by making it wiggly enough.
The converse is the more useful half. A fixed perimeter puts a firm ceiling on the area, and the shape that reaches it is the circle. Compare m of fencing:
- as a circle, m, enclosing
- as a square, side m, enclosing
- as a m by m rectangle,
Same fence, and the circle holds nearly four times what the thin rectangle does.
So the two measurements are not merely independent — they are lopsided. Perimeter limits area, but area does not limit perimeter. A farmer told only the length of the boundary wall knows the most land that could possibly be inside; a farmer told only the area knows nothing at all about how much wall it took.
This is why round shapes turn up wherever boundary is expensive and interior is valuable — a tank, a well, a chapati taking the most area from a given rim.
Take the m by m rectangle and cut a long thin notch into it from one side — say m deep and m wide. The area lost is only , but the boundary gains m and loses m, so the fence grows by m. **Almost no land lost, m of extra fence needed.**
Repeat with ten such notches and the fence passes m while the plot is still very nearly . There is no limit: a region of fixed area can be given as long a boundary as you like by making it wiggly enough.
The converse is the more useful half. A fixed perimeter puts a firm ceiling on the area, and the shape that reaches it is the circle. Compare m of fencing:
- as a circle, m, enclosing
- as a square, side m, enclosing
- as a m by m rectangle,
Same fence, and the circle holds nearly four times what the thin rectangle does.
So the two measurements are not merely independent — they are lopsided. Perimeter limits area, but area does not limit perimeter. A farmer told only the length of the boundary wall knows the most land that could possibly be inside; a farmer told only the area knows nothing at all about how much wall it took.
This is why round shapes turn up wherever boundary is expensive and interior is valuable — a tank, a well, a chapati taking the most area from a given rim.
Exam relevance
How do perimeter and circumference feed into JEE Main?
Because arc length is the bridge to radian measure, and radians are the language of every trigonometry and calculus chapter that follows.
This is the foundation for Class 11 Mathematics Trigonometric Functions, examined in JEE Main. The arc-length formula becomes the definition of the radian:
which is the Class 9 formula with the clumsy replaced by a unit chosen to make the factor disappear. A full turn is radians precisely because the whole circumference is — so the conversion radians is not an arbitrary fact but a restatement of this page. **A student who has computed a arc as one sixth of cm already understands radians and has not been told the name.
Circular motion depends on the same formula.** Class 11 Physics Motion in a Plane, examined in JEE Main and NEET, uses and the distance travelled along a circular path, both of which are arc length with time brought in. The wheel-and-revolutions calculation here is the same arithmetic that later gives the speed of a rotating body.
The maximisation in the previous section is treated properly in Class 12 Application of Derivatives, where fixed-perimeter and fixed-area problems are standard. The Class 9 observation that the circle encloses the most for a given boundary is the result those questions verify with calculus.
Where composite figures lead. Class 12 Application of Integrals finds areas bounded by curves, and the habit built here — decompose the boundary, handle each piece, add — is the same strategy with integration replacing the formulas. The decomposition is the transferable skill; the formulas are not.
What the questions look like. For board work, expect perimeter of a composite figure, circumference from radius or radius from circumference, arc length and sector perimeter for a given angle, and a running-track or fencing-cost problem — all with the formula, substitution and units shown. For JEE Main, the direct appearance is inside trigonometry and circular-motion questions, never as a perimeter question on its own.
How board and competitive emphasis differ. A board paper rewards the **stated value of **, the substitution written out, and the units on the answer. A competitive paper works in radians and exact multiples of , so disappears entirely — an answer there is more likely to read cm than cm.
The single trap that costs the most marks. Giving the arc length when the question asked for the sector's perimeter, or leaving out the diameter of a semicircular region. Both come from treating the curved formula as the whole answer, when the boundary also runs along straight edges — , not . Walking the boundary with a finger catches every instance of it, and it is the one habit from this page worth keeping for the next three years.
This is the foundation for Class 11 Mathematics Trigonometric Functions, examined in JEE Main. The arc-length formula becomes the definition of the radian:
which is the Class 9 formula with the clumsy replaced by a unit chosen to make the factor disappear. A full turn is radians precisely because the whole circumference is — so the conversion radians is not an arbitrary fact but a restatement of this page. **A student who has computed a arc as one sixth of cm already understands radians and has not been told the name.
Circular motion depends on the same formula.** Class 11 Physics Motion in a Plane, examined in JEE Main and NEET, uses and the distance travelled along a circular path, both of which are arc length with time brought in. The wheel-and-revolutions calculation here is the same arithmetic that later gives the speed of a rotating body.
The maximisation in the previous section is treated properly in Class 12 Application of Derivatives, where fixed-perimeter and fixed-area problems are standard. The Class 9 observation that the circle encloses the most for a given boundary is the result those questions verify with calculus.
Where composite figures lead. Class 12 Application of Integrals finds areas bounded by curves, and the habit built here — decompose the boundary, handle each piece, add — is the same strategy with integration replacing the formulas. The decomposition is the transferable skill; the formulas are not.
What the questions look like. For board work, expect perimeter of a composite figure, circumference from radius or radius from circumference, arc length and sector perimeter for a given angle, and a running-track or fencing-cost problem — all with the formula, substitution and units shown. For JEE Main, the direct appearance is inside trigonometry and circular-motion questions, never as a perimeter question on its own.
How board and competitive emphasis differ. A board paper rewards the **stated value of **, the substitution written out, and the units on the answer. A competitive paper works in radians and exact multiples of , so disappears entirely — an answer there is more likely to read cm than cm.
The single trap that costs the most marks. Giving the arc length when the question asked for the sector's perimeter, or leaving out the diameter of a semicircular region. Both come from treating the curved formula as the whole answer, when the boundary also runs along straight edges — , not . Walking the boundary with a finger catches every instance of it, and it is the one habit from this page worth keeping for the next three years.
Key takeaways
Perimeter, circumference, arcs and sectors: quick revision
- Perimeter is the sum of every boundary edge, counted once. For a regular polygon, — a hexagon of side cm gives cm.
- An irregular five-sided plot of m has m; a quadrilateral of perimeter cm with sides has a fourth side of cm.
- Cutting a square out of a CORNER does not change the perimeter: a m plot stays at m, since the two inward edges replace two equal ones.
- A notch in the MIDDLE of a side does change it: an m room goes from m to m.
- Never include an internal line in a perimeter, and walk the boundary instead of reaching for .
- ****, and . Take for radii that are multiples of , else , and say which you used.
- cm gives cm; cm gives cm; m gives m; cm gives cm.
- A wheel of radius cm has cm m, so revolutions cover ** m.
- is the same ratio for every circle**, and is only an approximation to it.
- **A semicircular region's perimeter is ** — for cm, cm.
- **Arc length ; sector perimeter arc **.
- Radius cm with : arc cm, sector perimeter cm. Radius cm with : arc cm, perimeter cm.
- **The semicircle is the case** of the sector formula.
- An arc of cm on a radius- circle gives ; a arc of cm gives cm.
- A sector lawn of radius m has arc m and needs ** m of fencing — the arc and the sector perimeter are different answers.
- A running track** of two m straights and two semicircular ends of diameter m: the ends make one circle, m, so m. At /m the fence costs **.
- Two equal semicircular ends make one full circle** — use directly.
- A cm square with a corner quadrant of radius cm removed has cm.
- A side that a semicircle is attached along stops being boundary.
- Perimeter limits area but area does not limit perimeter: m of fence encloses as a circle, as a square, and only as a m rectangle.
Measure something round at home with a thread, divide its circumference by its diameter, and see how close you get to — the gap is your measuring error, not the circle's.
- An irregular five-sided plot of m has m; a quadrilateral of perimeter cm with sides has a fourth side of cm.
- Cutting a square out of a CORNER does not change the perimeter: a m plot stays at m, since the two inward edges replace two equal ones.
- A notch in the MIDDLE of a side does change it: an m room goes from m to m.
- Never include an internal line in a perimeter, and walk the boundary instead of reaching for .
- ****, and . Take for radii that are multiples of , else , and say which you used.
- cm gives cm; cm gives cm; m gives m; cm gives cm.
- A wheel of radius cm has cm m, so revolutions cover ** m.
- is the same ratio for every circle**, and is only an approximation to it.
- **A semicircular region's perimeter is ** — for cm, cm.
- **Arc length ; sector perimeter arc **.
- Radius cm with : arc cm, sector perimeter cm. Radius cm with : arc cm, perimeter cm.
- **The semicircle is the case** of the sector formula.
- An arc of cm on a radius- circle gives ; a arc of cm gives cm.
- A sector lawn of radius m has arc m and needs ** m of fencing — the arc and the sector perimeter are different answers.
- A running track** of two m straights and two semicircular ends of diameter m: the ends make one circle, m, so m. At /m the fence costs **.
- Two equal semicircular ends make one full circle** — use directly.
- A cm square with a corner quadrant of radius cm removed has cm.
- A side that a semicircle is attached along stops being boundary.
- Perimeter limits area but area does not limit perimeter: m of fence encloses as a circle, as a square, and only as a m rectangle.
Measure something round at home with a thread, divide its circumference by its diameter, and see how close you get to — the gap is your measuring error, not the circle's.