Question 1(a)
A rectangle has perimeter 14 cm and breadth 2 cm. Find its length.
Steps only
P=2(l+b).
14=2(l+2), so l+2=7.
l=5 cm.
Answer
The length is 5 cm.
CLASS 6 · GANITA PRAKASH · CHAPTER 6
Perimeter and Area
Understand boundary length, area, composite shapes, and how perimeter and area vary.
77 worked answersAll 8 end-of-chapter questions
What you will learn
- Find polygon perimeters and solve boundary-length problems.
- Measure rectangular and composite areas in square units.
- Explain how triangle area and shape rearrangements relate to area and perimeter.
77 worked answers · short steps first, explanations when you need them
6.1 Perimeter
Textbook pages 131–136
Question 1(b)
A square has perimeter 20 cm. Find its side length.
Steps only
All four sides are equal.
20÷4=5 cm.
Answer
Each side is 5 cm.
Question 1(c)
A rectangle has perimeter 12 m and length 3 m. Find its breadth.
Steps only
12=2(3+b), so 6=3+b.
b=3 m.
Answer
The breadth is 3 m.
Question 2
A wire forms a 5 cm by 3 cm rectangle. It is reshaped into a square. Find the square’s side length.
Steps only
Wire length=2(5+3)=16 cm.
Divide the same 16 cm equally among four sides: 16÷4=4 cm.
Answer
The square side is 4 cm.
Question 3
A triangle has perimeter 55 cm. Two sides are 20 cm and 14 cm. Find the third side.
Steps only
Add known sides: 20+14=34 cm.
Subtract from the perimeter: 55−34=21 cm.
Answer
The third side is 21 cm.
Question 4
A rectangular park is 150 m long and 120 m wide. Fencing costs ₹40 per metre. Find the cost.
Steps only
Perimeter=2(150+120)=540 m.
Cost=540×₹40=₹21,600.
Answer
The fencing costs ₹21,600.
Question 5(a)
A 36 cm string forms a square. Find each side.
Steps only
There are 4 equal sides.
36÷4=9 cm.
Answer
Each side is 9 cm.
Question 5(b)
A 36 cm string forms an equilateral triangle. Find each side.
Steps only
There are 3 equal sides.
36÷3=12 cm.
Answer
Each side is 12 cm.
Question 5(c)
A 36 cm string forms a regular hexagon. Find each side.
Steps only
A hexagon has 6 equal sides.
36÷6=6 cm.
Answer
Each side is 6 cm.
Question 6
A 230 m by 160 m field gets three complete rounds of rope. How much rope is needed?
Steps only
One perimeter=2(230+160)=780 m.
Three rounds=3×780=2340 m.
Answer
2340 m of rope.
Question track-1
Akshi runs five rounds of the outer rectangular track, 70 m by 40 m. How far does she run?
Steps only
One round=2(70+40)=220 m.
Five rounds=5×220=1100 m.
Answer
Akshi runs 1100 m.
Question track-2
Toshi runs seven rounds of the inner track, 60 m by 30 m. Who runs farther than Akshi’s five outer rounds?
Steps only
Inner perimeter=2(60+30)=180 m.
Toshi runs 7×180=1260 m.
Compare 1260 m with Akshi’s 1100 m.
Answer
Toshi runs farther: 1260 m compared with 1100 m.
Question track-3(a)
Akshi starts at a corner of a 70 m by 40 m track and runs 250 m. How can her position be marked?
Steps only
One round is 220 m, leaving 250−220=30 m after one full round.
From the start, follow the track 30 m along its first 70 m side.
Answer
Mark A 30 m along the first 70 m side from the start.
Question track-3(b)
Where is Akshi after 500 m?
Steps only
500=2×220+60.
After two laps, move 60 m from the start along the next partial lap.
Answer
She is 60 m from the start along the first side.
Question track-3(c)
How many full rounds has Akshi completed after 1000 m?
Steps only
1000÷220=4 remainder 120.
Answer
Four full rounds, with 120 m into the next round.
Question track-3(d)
Toshi runs 250 m on a 60 m by 30 m inner track. How can her position be marked?
Steps only
One round is 180 m, leaving 250−180=70 m.
From her marked start and direction, trace 60 m along the first side and 10 m along the next.
Answer
Mark X after one full lap plus 70 m: 60 m along the first side and 10 m along the next.
Question track-3(e)
Where is Toshi after 500 m?
Steps only
500=2×180+140.
After two laps, trace 140 m from her start along the track.
Answer
She has completed two rounds and is 140 m into the next.
Question track-3(f)
How many complete rounds has Toshi run after 1000 m?
Steps only
1000÷180=5 remainder 100.
Answer
Five complete rounds, with 100 m of the next round.
Question deep-dive
For 350 m races on square tracks of side 100 m and 150 m, how far before the finish should each runner start?
Steps only
Inner perimeter=400 m, so run 350 m by starting 50 m before the finish.
Outer perimeter=600 m; 350 m can be made as 125+150+75 along the shown route.
Answer
Inner start is 50 m before the finish along its route; outer start follows the diagram’s 125 m, 150 m and 75 m segments.
Question diagonal-triangle
A triangle uses three straight grid sides of 3 units each. Is its perimeter 9 units if one side is diagonal?
Steps only
The diagonal crosses a unit square from corner to corner.
That diagonal is longer than one straight unit, so replacing it by the same count of straight units underestimates the boundary.
Answer
A square’s diagonal is longer than either side, so three side-counts of three straight units cannot equal the true boundary length.
Question s-and-d
How should a grid-shape perimeter with straight and diagonal edges be recorded?
Steps only
Count straight unit edges as s and diagonal edges as d.
Add the counts separately.
Answer
The four pictured shapes have perimeters 8s+2d, 4s+6d, 12s+6d and 18s+6d.
Question regular-polygon
How do you find the perimeter of a regular polygon?
Steps only
Each side has the same length.
Multiply the number of sides by one side length.
Answer
Perimeter = number of sides × side length.
Question split-rejoin-b
A 6 cm by 4 cm chit is cut into two 6 cm by 2 cm pieces. Find the perimeter of arrangement b.
Steps only
Trace the outside boundary once: 6+2+6+2+4+6+2=28 cm.
Do not count the joined edge inside the L-shape.
Answer
Arrangement b has perimeter 28 cm.
Question split-rejoin-c
Find the perimeter of the T-shaped arrangement c made from the two equal pieces.
Steps only
Follow the outside outline: 2+6+2+2+6+2+6+2=28 cm.
Answer
Arrangement c has perimeter 28 cm.
Question split-rejoin-d
Find the perimeter of arrangement d, where the pieces overlap along a 3 cm segment.
Steps only
The exposed boundary segments add to 6+2+3+2+6+2+3+2.
Their sum is 26 cm.
Answer
Arrangement d has perimeter 26 cm.
Question split-rejoin-22
How can the two 6 cm by 2 cm pieces be joined to make a shape with perimeter 22 cm?
Steps only
Each piece alone has perimeter 16 cm, so together before joining they contribute 32 cm.
Join them along a 5 cm portion of one long side. This hides 5 cm from each outline.
32−2×5=22 cm.
Answer
Join the pieces along 5 cm of their long edges, leaving 1 cm of that edge exposed at each end. The resulting perimeter is 22 cm.
Perimeter examples and explorations
Textbook pages 131–136
Question tablecloth-example
A 3 m by 2 m rectangular tablecloth needs lace all around its edge. How much lace is needed?
Steps only
P=2(3+2)=2×5=10 m.
Answer
10 m of lace is needed.
Question square-park-example
Usha runs 3 rounds of a square park with side 75 m. How far does she run?
Steps only
One lap=4×75=300 m.
Three laps=3×300=900 m.
Answer
Usha runs 900 m.
Question regular-similarity
What is similar about a square and an equilateral triangle?
Steps only
Both have all sides equal, and both are regular polygons.
Their angles are different: 90° in a square and 60° in an equilateral triangle.
Answer
A square and an equilateral triangle each have equal sides and equal angles; their angles are 90° and 60° respectively.
Question regular-objects
How can you find and generalise the perimeter of regular polygons around you?
Steps only
Measure one side of a regular object and count its sides.
Multiply: perimeter = number of sides × side length.
For example, a regular hexagon with side 4 cm has perimeter 6×4=24 cm.
Answer
For n equal sides of length s, P=n×s. Examples from the surroundings vary.
Question cut-paper-estimate
How can you estimate and then check the perimeter of an irregular shape cut from paper?
Steps only
Trace or cut out the shape. Estimate the outside boundary in shorter sections.
Measure straight edges with a ruler; use a thread for curved edges, then measure the thread.
Add the measured sections and compare with your estimate.
Answer
Add the lengths around the boundary; compare this measured total with your estimate.
6.2 Area
Textbook pages 137–141
Question formula
What are the area formulas for a rectangle and a square?
Steps only
Rectangle: multiply length by width.
A square has equal length and width, so multiply side by itself.
Answer
Rectangle area = length×width; square area = side×side.
Question example-floor
A 5 m by 4 m floor has a 3 m square carpet. Find the uncovered area.
Steps only
Floor area=5×4=20 m².
Carpet area=3×3=9 m².
Uncovered=20−9=11 m².
Answer
11 m².
Question example-flowerbeds
A 12 m by 10 m plot has four square flower beds of side 4 m. Find the remaining area.
Steps only
Plot area=12×10=120 m².
Each bed=4×4=16 m²; four beds=64 m².
Remaining area=120−64=56 m².
Answer
56 m².
Question 1
A rectangular garden has area 300 m² and length 25 m. Find its width.
Steps only
Area=length×width.
Width=300÷25=12 m.
Answer
12 m.
Question 2
A 500 m by 200 m plot is tiled at ₹8 per 100 m². Find the cost.
Steps only
Area=500×200=100,000 m².
There are 100,000÷100=1000 groups of 100 m².
Cost=1000×₹8=₹8000.
Answer
₹8,000.
Question 3
A 100 m by 50 m coconut grove needs 25 m² per tree. What is the maximum number of trees?
Steps only
Grove area=100×50=5000 m².
5000÷25=200.
Answer
200 trees.
Question 4
How can you find the area of an irregular rectilinear figure by splitting it into rectangles?
Steps only
Draw cuts that divide the figure into non-overlapping rectangles.
Find each rectangle’s length×width and add the areas.
Answer
For the two pictured figures, the areas are 28 m² and 9 m² respectively.
Question grid-area
How do you estimate a shape’s area using a one-unit square grid?
Steps only
Count each full square as 1 square unit.
Ignore pieces under half a square; count pieces over half as 1; count an exact half as 1/2.
Answer
Add the counted full and estimated partial squares.
Question area-shapes
What are the areas of the four grid figures in the chapter?
Steps only
Count full and partial unit squares according to the stated convention.
Answer
Area is a count of square units covering the region; include half-squares as specified.
Question rectangles-24
Among whole-unit rectangles with area 24 square units, which have the greatest and least perimeter?
Steps only
Factor pairs: 1×24, 2×12, 3×8, 4×6.
Perimeters: 50, 28, 22, 20 units.
Answer
For each factor pair use P=2(l+w).
Question rectangles-32
Among whole-unit rectangles with area 32 square units, which have the greatest and least perimeter?
Steps only
Factor pairs: 1×32, 2×16, 4×8.
Perimeters: 66, 36, 24 units.
Answer
Compare the perimeter for each factor pair.
Question rectangle-area-rule
For a fixed whole-number area, which rectangle has the greatest perimeter and which has the least?
Steps only
For area N, a 1×N rectangle has perimeter 2(N+1), the greatest among whole-number factor pairs.
For the least perimeter, compare factor pairs and choose the pair closest together.
For example, area 24: 1×24 gives 50 (greatest), while 4×6 gives 20 (least).
Answer
With whole-number side lengths and fixed area, a 1-by-N rectangle is longest and has the greatest perimeter. The most balanced factor pair gives the least perimeter.
Question circle-units
Why are squares convenient units for measuring area compared with circles?
Steps only
Equal squares fit together edge-to-edge without leaving gaps or overlapping.
Circles leave gaps when packed.
Answer
This makes square units useful for measuring area.
Area estimates and measurement
Textbook pages 140–141
Question compare-figures
Two outlined shapes look different. Which seems to enclose more area?
Steps only
A jagged or pointed outline can look larger even when its area is close to another shape.
Overlay each outline on a square grid and count whole and partial squares using the chapter’s rule.
Answer
A visual guess is uncertain; a grid estimate shows that the pictured shapes enclose approximately the same area.
Question circle-grid-estimate
A circle has diameter 3 units. How can its area be estimated using a unit-square grid?
Steps only
Trace the circle on the grid; count each full square as 1.
Ignore pieces under half a square, count pieces over half as 1, and exact halves as 1/2.
The estimate is about 7 square units; the exact grid count can vary slightly with placement.
Answer
This is an estimate, not an exact circle-area calculation. A diameter of 3 gives radius 1.5, so the region covers about 7 unit squares.
Question square-tiling-reasons
Why are squares more useful than circles for measuring a flat region?
Steps only
Same-sized squares fit edge-to-edge in rows with no gaps or overlaps.
Circles leave gaps, and different arrangements can fit different numbers into the same rectangle.
Answer
Squares tile the plane exactly, so they cover a region consistently and can be counted in rows and columns.
Question outside-corridor
The textbook floor plan gives the floor area outside its corridor. What is that area?
Steps only
Split the floor plan into its rectangular parts, or subtract the corridor area from the complete floor area.
Using the measurements shown in the plan gives 1200 m² outside the corridor.
Answer
The measured floor area outside the corridor is 1200 m².
Question school-playground
Find the area of the school playground shown in the textbook activity.
Steps only
Use the plan’s dimensions: length 100 m and width 90 m.
Area=100×90=9000 m².
Answer
The pictured playground occupies 9000 m².
Tangram areas
Textbook pages 139
Question 1
Which tangram pieces have equal areas?
Steps only
The two large triangles A and B match in size.
The two small triangles C and E also match in size.
Answer
A and B have equal areas; C and E have equal areas.
Question 2
How does the area of triangle D compare with triangle C?
Steps only
Triangle D can be covered by two copies of the small triangle C.
The same relation is shown by D covering C and E together.
Answer
D has twice the area of C: area(D)=area(C)+area(E)=2 area(C).
Question 3
How does the tangram square F compare in area with triangle D?
Steps only
Rearrange the square F to cover triangle D, or compare their unit-triangle pieces.
Answer
F and D have equal areas.
Question 4
How does parallelogram G compare in area with square F?
Steps only
Both pieces can be assembled from the same two small-triangle units.
Answer
G and F have equal areas.
Question 5
How does large triangle A compare with parallelogram G?
Steps only
A is four small-triangle units; G is two such units.
So A has twice G’s area.
Answer
Area(A)=2 area(G).
Question 6
If small triangle C has area 1 square unit, what is the area of all seven pieces together?
Steps only
A and B each have area 4.
C and E each have area 1; D, F and G each have area 2.
Add: 4+4+1+1+2+2+2=16.
Answer
The complete tangram has area 16 square units.
Question 7
If all seven tangram pieces are rearranged into a rectangle, how does its area compare with the original square?
Steps only
Rearranging does not add or remove any pieces.
The rectangle contains exactly the same seven pieces.
Answer
The rectangle and original square have equal area: 16 square units when C is one unit.
Question 8
When the tangram pieces are rearranged into a rectangle, must its perimeter equal the square’s perimeter?
Steps only
The same pieces preserve total area.
Perimeter measures only the outside edges, which change when the pieces are rearranged.
Answer
No. The areas stay equal, but the rectangle’s perimeter need not equal the square’s perimeter.
6.3 Area of a Triangle
Textbook pages 142–149
Question diagonal
What is the relationship between a rectangle’s diagonal and the two triangles it forms?
Steps only
A diagonal splits the rectangle into two matching triangles.
The two equal halves have equal area.
Answer
Each triangle has half the rectangle’s area.
Question grid-triangles
On the textbook grid, find the areas of blue triangle BAD and red triangle ABE.
Steps only
The rectangle ABCD measures 5 units by 4 units, so its area is 5×4=20 square units.
Triangle BAD is half the rectangle: 20÷2=10.
Triangle ABE has the same base 5 and height 4: (5×4)÷2=10.
Answer
The two triangles look different, but both have area 10 square units.
Question grid-triangle-conclusion
What do triangles BAD and ABE show about triangles drawn inside a rectangle?
Steps only
BAD is half of rectangle ABCD.
ABE can be split into two smaller triangles, each half of one smaller rectangle; together those two rectangles fill ABCD.
Answer
Both triangles have half the area of rectangle ABCD, even though their shapes look different.
Question any-triangle
Why is a triangle’s area one-half of base times perpendicular height?
Steps only
A matching parallelogram or rectangle can be split into two equal-area copies of the triangle.
The full rectangle area is base×height.
One copy has half that area.
Answer
The triangle occupies half the area of a rectangle/parallelogram with the same base and perpendicular height.
Question 1
How do you find the areas of the five figures made from rectangles and triangles?
Steps only
Split each outline into familiar pieces.
For each rectangle use l×w; for each triangle use (b×h)÷2.
Add the non-overlapping parts.
Answer
Each figure is the sum of its rectangle and triangle parts; keep units squared.
Question 2
Using nine unit squares joined edge-to-edge, find the least and greatest possible perimeter.
Steps only
A 3×3 square has perimeter 4×3=12 units.
A 1×9 row has perimeter 2(1+9)=20 units.
Answer
Least 12 units; greatest 20 units.
Question 3
Can nine unit squares make perimeter 18? Give an example.
Steps only
For nine squares, start with 36 unit edges.
A connected arrangement with nine shared sides has 36−2×9=18 exposed edges.
Answer
Yes. Arrange nine squares with nine shared sides; the outside boundary is 18 units.
Question 4
Can different nine-square shapes have the same perimeter?
Steps only
Perimeter depends on how many square sides are shared, not only on the outline’s overall form.
Different arrangements can have the same number of shared edges.
Answer
Yes. More than one arrangement can have a given perimeter; the compact minimum arrangement is the 3×3 square.
Question 5
When does attaching one unit square increase, decrease, or preserve a shape’s perimeter?
Steps only
If the new square shares one edge, the perimeter increases by 2.
If it shares two edges, two old edges are covered and two new outside edges appear, so the perimeter stays the same.
If it shares three edges, the perimeter decreases by 2.
Answer
One shared side: +2; two shared sides: no change; three shared sides: −2 units.
Question charan-house
Find the missing room sizes in Charan’s plan and the house area.
Steps only
Use aligned room walls and the 35 ft by 30 ft outer dimensions to infer missing room sides.
Multiply the outer length by width: 35×30.
Answer
The footprint area is 1050 sq ft. The plan gives small bedroom 15×12, utility 15×3, hall 20×12, parking 15×3 and garden 20×3.
Question sharan-house
What is Sharan’s house area, and how do its area and perimeter compare with Charan’s?
Steps only
Sharan’s outer dimensions are 42 ft by 25 ft.
Area=42×25=1050 sq ft.
Perimeter=2(42+25)=134 ft. Charan’s 35×30 perimeter is 130 ft.
Answer
Both houses have area 1050 sq ft, but Sharan’s perimeter is 134 ft and Charan’s is 130 ft.
Question maze-1
How can a missing area in the area maze be found?
Steps only
Use that a rectangle’s area is length×width.
Subtract the known adjoining region from the whole area.
Answer
For the four pictured puzzles, missing answers are 30 cm², 9 cm², 16 cm² and 5 cm respectively.
Question 1
Give dimensions for a rectangle whose area equals the combined areas of a 5 m×10 m rectangle and a 2 m×7 m rectangle.
Steps only
Areas are 50 m² and 14 m².
Total=64 m².
Factor pairs include 16×4, 32×2 and 8×8.
Answer
Examples: 16 m×4 m, 32 m×2 m or 8 m×8 m.
Question 2
A garden has area 1000 m² and length 50 m. Find its width.
Steps only
Width=area÷length=1000÷50.
Answer
20 m.
Question 3
A room is 5 m by 4 m. A 3 m square carpet covers part of it. Find the uncovered area.
Steps only
Floor area=20 m².
Carpet area=9 m².
Uncovered=20−9.
Answer
11 m².
Question 4
A 15 m by 12 m garden has four flower beds, each 2 m by 1 m. Find the lawn area.
Steps only
Garden area=15×12=180 m².
Beds area=4×(2×1)=8 m².
Lawn=180−8.
Answer
172 m².
Question 5
Shape A has area 18 square units and Shape B has area 20 square units. Give examples where A has longer perimeter.
Steps only
Choose A as 2×9: P=22; B as 5×4: P=18.
Both areas meet the conditions and 22>18.
Answer
For example, A can be 2×9 (area 18, perimeter 22) and B can be 5×4 (area 20, perimeter 18).
Question 6
On a page, draw a rectangular border 1 cm from the top and bottom and 1.5 cm from the left and right sides. What is its perimeter?
Steps only
Measure the page height H and width W.
The inner border measures (H−2) cm by (W−3) cm.
Its perimeter is 2[(H−2)+(W−3)] cm.
Answer
Measure your own page, subtract 2 cm from its height and 3 cm from its width, then double the sum. For a 20 cm by 15 cm page, the border is 18 cm by 12 cm, so its perimeter is 60 cm.
Question 7
Inside a 12-unit by 8-unit rectangle, draw a smaller rectangle with half its area.
Steps only
Outer area=12×8=96 square units.
Half is 48.
An inner 8×6 rectangle has area 48 and fits with margins.
Answer
For example, draw an 8×6 rectangle inside; its area is 48 square units.
Question 8
A square is folded in half and cut along the fold to make two rectangles. Which perimeter statement is always true?
Steps only
Let the square side be s. Each rectangle is s by s/2.
Their combined perimeters are 2×2(s+s/2)=6s.
The square perimeter is 4s, and 6s=1.5×4s.
Answer
The two rectangle perimeters together equal 1½ times the square’s perimeter (choice c).