Perimeter and Area | CBSE Class 6 Maths Notes

This note covers perimeter, rectangles, squares, triangles, regular polygons, missing lengths, boundary problems, area, square units, grid estimates, triangle areas, rearranging shapes, tangrams, house plans and area puzzles.

What is perimeter, and how do we find it for a rectangle?

Definition: The perimeter of a closed plane figure is the distance travelled along its boundary once. A closed plane figure is a flat shape whose boundary encloses a region. Its area measures the region enclosed.

A polygon is a closed plane figure made from line segments, which are straight parts joining two endpoints. These segments form its sides. To find its perimeter, add the lengths of all the sides on the outer boundary.

The words length and breadth, or width, describe the two side measurements of a rectangle. An angle is the opening between two sides meeting at a corner. A rectangle has four sides and four right angles, or square corners. Its opposite sides are equal in length.

Result: Perimeter of a rectangle

Perimeter of a rectangle = 2 × (length + breadth). Here, = means “is equal to”, + means addition and × means multiplication. The brackets group the measurements that must be added before multiplying by two.

This rule follows from adding length, breadth, length and breadth. Each measurement occurs twice. A sum is the result of addition. Adding just one length and one breadth takes us along two sides, rather than around the whole rectangle.

Worked example 1. Find the perimeter of a rectangle with length 12 cm and breadth 8 cm. The abbreviation cm means centimetres, a unit for measuring length.

Answer: Perimeter = 12 + 8 + 12 + 8 = 40 cm. Using the shorter rule gives 2 × (12 + 8) = 2 × 20 = 40 cm.

The rectangle has corners labelled A, B, C and D, in order around its boundary. The top side is labelled 12 cm and the left side 8 cm.

Reference: NCERT Class 6, page 129

The letters name the corners; a pair of letters names the side joining them. Thus, AB is the side from A to B. The full journey is AB + BC + CD + DA. Starting at a different corner does not change which four sides are counted.

How do squares, triangles and regular polygons simplify perimeter?

A square has four equal sides and four right angles. Because its sides have equal lengths, we can multiply one side length by four instead of writing the same length four times. “Quadruple” means four times a quantity.

Result: Perimeter of a square

Perimeter of a square = 4 × length of a side. This is a boundary measurement. It tells us the length needed to go once around the square, rather than the amount of space inside it.

Worked example 2. Debojeet puts coloured tape around a square photo frame whose side is 1 m. The abbreviation m means metres, another unit of length. How much tape is required?

Answer: The tape follows all four sides. Its length is 1 + 1 + 1 + 1 = 4 m, or 4 × 1 = 4 m.

A triangle is a polygon with three sides. Its perimeter is the sum of those three side lengths. Equal sides are not assumed merely because a figure is a triangle.

Worked example 3. Find the perimeter of a triangle with side lengths 4 cm, 5 cm and 7 cm.

Answer: Add all three lengths: perimeter = 4 + 5 + 7 = 16 cm. The answer measures the complete boundary.

Result: Perimeter of a regular polygon

A regular polygon has all its sides equal and all its angles equal. An equilateral triangle has three equal sides and three equal angles; a regular pentagon has five of each.

Perimeter of a regular polygon = number of sides × length of a side. For an equilateral triangle, this becomes three times a side. The square and equilateral triangle therefore use the same idea: repeated addition of equal lengths becomes multiplication.

How can we recover missing lengths or reshape a wire?

A known perimeter tells us the total length around a figure. To find a missing side, work backwards from that total using the relationship between the sides. The method depends on whether the figure is a rectangle, square or triangle.

How do we reverse a perimeter calculation?

1. Identify the figure and write its perimeter rule in words.
2. For a rectangle, halve the perimeter to obtain length plus breadth.
3. Subtract the known measurement to obtain the missing measurement.
4. Substitute the answer into the original perimeter rule to check it.

Worked example 4. A rectangle has perimeter 14 cm and breadth 2 cm. Find its length.

Answer: Length plus breadth = 14 ÷ 2 = 7 cm. Here, ÷ means division. Length = 7 − 2 = 5 cm, where − means subtraction. Check: 2 × (5 + 2) = 14 cm.

For a square with perimeter 20 cm, divide the perimeter by four to find a side of 5 cm. For a triangle with perimeter 55 cm and two sides of 20 cm and 14 cm, subtract both known sides to obtain the third side, 21 cm.

What stays the same when a wire is bent again?

Worked example 5. A wire forms a rectangle with sides 5 cm and 3 cm. It is straightened and bent into a square. Find the square's side.

Answer: The wire length is the rectangle's perimeter, 2 × (5 + 3) = 16 cm. The square uses this same boundary length. Each of its four equal sides measures 16 ÷ 4 = 4 cm.

The new shape changes the arrangement of the wire, while its total length remains the same. The quantity to preserve is perimeter. Do not carry one of the rectangle's side lengths directly into the square: the new sides must share the whole wire equally.

A similar problem uses a 36 cm string. A square made from it has sides of 9 cm, an equal-sided triangle has sides of 12 cm, and an equal-sided hexagon, a six-sided polygon, has sides of 6 cm.

How do perimeter calculations solve practical boundary problems?

Lace around a tablecloth, fencing around a field and a complete round of a track all involve boundary length. First calculate the perimeter of the shape. Then decide whether the problem needs one boundary, several rounds or a cost based on that length.

Worked example 6. Akshi wants lace around a rectangular tablecloth 3 m long and 2 m wide. Find the lace length.

Answer: Lace follows the tablecloth's perimeter. Required length = 2 × (3 + 2) = 2 × 5 = 10 m.

Worked example 7. Usha takes three rounds of a square park with side 75 m. Find the total distance travelled.

Answer: One round = 4 × 75 = 300 m. Three rounds cover 3 × 300 = 900 m. The number of rounds multiplies the perimeter.

How do rounds and rates affect the answer?

A rate describes an amount for each unit, such as the cost for each metre of fencing. The symbol ₹ means rupees. For a rectangular park 150 m long and 120 m wide, the perimeter is 540 m.

At ₹40 per metre, fencing that park costs 540 × 40 = ₹21,600. A farmer's field measuring 230 m by 160 m has perimeter 780 m. Three rounds of rope around it require 3 × 780 = 2340 m.

Toshi travels farther even though the inner track is smaller. Comparing total distances requires both the length of one round and the number of rounds. Comparing the track perimeters alone would leave out part of the problem.

Note: In races, usually there is a common finish line for all the runners. When tracks have different lengths, starting positions can be chosen so that the runners cover the required race distance before reaching that line.

Why must straight and diagonal grid lengths be counted separately?

A grid arranges points or lines in regular rows and columns. On a square grid, a step along a square's side and a step across its diagonal are different lengths. A diagonal joins two corners of a polygon that are not next to each other.

In the dot-grid triangle, the red boundary parts follow horizontal and vertical grid directions. The blue boundary follows a sloping direction across the grid squares. Counting each part as the same unit would mix two unequal lengths.

A triangle on a dot grid has red horizontal and vertical sides and a blue sloping side. The red sides together contain six straight intervals; the blue side contains three diagonal intervals.

Reference: NCERT Class 6, pages 134 to 135

How is the mixed perimeter written?

Let s represent one straight interval's length and d one diagonal interval's length, both measured using the same chosen length unit. The triangle's perimeter is 6s + 3d units. Writing 6s means six times s; 3d means three times d.

The diagonal interval is longer than the straight interval. Therefore, this triangle's perimeter is more than nine straight units. Its boundary has nine intervals, but these are not nine equal-length intervals.

Keep the straight and diagonal contributions separate until their lengths are known. This also explains why measuring a red segment and a blue segment is useful: it checks the assumption behind the counting method. A count becomes a length measurement only when the unit being counted is clear.

What does area measure, and how do rectangle and square formulas work?

Definition: Area is the amount of region enclosed by a closed figure. A square unit is the area of a square measuring one length unit on each side. Area is generally measured in square units.

For example, m² means square metres, the same as “sq m”. Similarly, cm² means square centimetres. The raised ² in these unit symbols indicates square measure. A length in metres and an area in square metres describe different quantities.

Result: Areas of rectangles and squares

Area of a rectangle = length × width. Square grid paper helps explain this rule by arranging equal unit squares in rows. Counting the squares across and the rows gives the total number covering the rectangular region.

Area of a square = side × side. The length and width of a square are equal, so the same measurement is multiplied by itself. These area rules count the space inside a shape, rather than adding its boundary lengths.

Worked example 8. A rectangular floor is 5 m long and 4 m wide. A square carpet with side 3 m is laid on it. Find the area not carpeted.

Answer: Floor area = 5 × 4 = 20 m². Carpet area = 3 × 3 = 9 m². Uncarpeted area = 20 − 9 = 11 m².

The remaining area is found by subtracting the covered part from the whole. Both quantities must be areas. Subtracting the carpet's side length from the floor's area would mix length and area and would not measure the uncovered region.

Worked example 9. Four square flower beds, each with side 4 m, occupy the four corners of land 12 m long and 10 m wide. Find the remaining area.

Answer: Whole land = 12 × 10 = 120 m². One bed = 4 × 4 = 16 m². Four beds = 4 × 16 = 64 m². Remaining land = 120 − 64 = 56 m².

The four beds are separate parts of the land. Finding one bed's area is an intermediate step, not the total area to remove. Multiply that area by the number of equal beds before subtracting from the whole land.

How can area help us find a width, a cost or a number of trees?

The rectangle rule can be used in reverse. If the area and one side length are known, divide the area by that side length to obtain the other measurement. The answer is a length, so it needs a length unit.

How do we find a missing width?

Worked example 10. A rectangular garden is 25 m long and has area 300 m². Find its width.

Answer: Width = area ÷ length = 300 ÷ 25 = 12 m. Check by multiplying: 25 × 12 = 300 m².

A second garden has length 50 m and area 1000 m². Its width is 1000 ÷ 50 = 20 m. In both cases, division reverses the multiplication used to calculate a rectangle's area.

How does an area rate work?

Worked example 11. A rectangular plot is 500 m long and 200 m wide. Tiling costs ₹8 per hundred square metres. Find the total cost.

Answer: Plot area = 500 × 200 = 100,000 m². There are 100,000 ÷ 100 = 1000 groups of one hundred square metres. Total cost = 1000 × 8 = ₹8000.

The phrase per hundred square metres matters. The price applies to a group of one hundred square metres, not to each square metre. Read the rate completely before deciding which multiplication or division to use.

A rectangular coconut grove is 100 m long and 50 m wide. Its area is 5000 m². If each tree requires 25 m², the maximum number of trees in this area calculation is 5000 ÷ 25 = 200.

These problems begin with the same area formula but use the result differently: one recovers a missing side, another counts priced groups of area, and another counts the area allocations for trees. State what each division is counting.

How can we estimate the area of an irregular shape?

An irregular shape need not have the equal sides and equal angles of a regular polygon. For a shape whose area is not directly found using a rectangle or square rule, squared paper provides a way to estimate its enclosed region.

An estimate is an approximate value. Trace the outline on transparent paper, then place it over squared or graph paper. Here, graph paper means paper ruled into a grid of small squares. Each small square represents one square unit.

Which counting conventions should we use?

1. Count each complete small square inside the region as one square unit.
2. Ignore a portion covering less than half a small square.
3. Count a portion covering more than half a small square as one square unit.
4. Count exactly half a small square as half a square unit.

These conventions give a consistent way to deal with boundary squares. They do not turn every curved or irregular boundary into an exact count. Areas can be estimated, or even determined exactly, by breaking regions into suitable squares, rectangles and triangles.

Why are squares useful area units?

Unit squares can cover a rectangular region without gaps or overlaps. Equal circles leave gaps when packed together, making an accurate area measurement difficult. Different arrangements of circles can also give different counts inside the same rectangle.

Two rectangular outlines contain rows of equal circles. In one arrangement the circles line up in columns; in the other they are staggered. The two arrangements contain 42 and 44 circles respectively.

Reference: NCERT Class 6, page 141

Area is generally measured using squares. Rectangles and triangles are also useful for dividing a larger region into calculable parts. The aim is to cover the region accurately, with each part accounted for and without counting overlapping pieces twice.

How do rectangles help us understand the area of a triangle?

Draw a rectangle and one of its diagonals, then cut along the diagonal. The two resulting triangles fit exactly over each other. They therefore have equal areas, and together they make the whole rectangle.

Result: A diagonal divides a rectangle into equal-area triangles

Each triangle has half the area of the rectangle. This conclusion comes from comparing the pieces, not from judging how large a sloping side looks. The same cut-and-compare activity can be tried with rectangles of different dimensions and with a square.

Rectangle ABCD has A at the lower left, B at the lower right, C at the upper right and D at the upper left. Point E lies on the top side; F lies directly below E on the bottom side. Blue lines outline triangle BAD, and red lines outline triangle ABE.

Reference: NCERT Class 6, page 143

The letters in each triangle's name identify its three corners. The segment EF joins E to F and splits the rectangle into the smaller rectangles AFED and BFEC. It also splits triangle ABE into triangles AEF and BEF.

Why do the blue and red triangles have equal areas?

1. Triangle BAD is one of the two triangles formed by the diagonal from D to B, so it has half the area of rectangle ABCD.
2. Triangle AEF has half the area of the smaller rectangle AFED.
3. Triangle BEF has half the area of the smaller rectangle BFEC.
4. Adding the two smaller triangle areas gives half the combined area of the two smaller rectangles, which is half the area of ABCD.

Thus, triangles BAD and ABE have the same area although they look different. Dividing a figure into rectangles and triangles helps explain the area relationship. When adding parts, use pieces that together cover the required region without overlaps or gaps.

Can equal areas have different perimeters?

Area and perimeter are different measurements. Two closed figures can have the same area with different perimeters, or the same perimeter with different areas. Rearranging pieces can preserve the amount of region while changing the shape of its outer boundary.

One figure places nine equal unit squares in a solid square. The other places nine unit squares in a C-shaped arrangement. Their areas are both nine square units, while their perimeters are 12 and 20 units respectively.

Reference: NCERT Class 6, page 145

A connected figure has its pieces joined together. In the nine-square activity, every square must share at least one whole side with another square, all the squares must form one connected figure, and there must be no holes.

What happens at a shared edge?

When two squares meet along a whole side, that joined side is inside the combined shape. It does not belong to the outer boundary. Count the exposed edges when finding the perimeter of the combined figure.

This explains how the same collection of unit squares can have different perimeters. More of their edges may lie inside one arrangement than another. The area still counts the nine unit squares, while the perimeter counts the boundary that remains exposed.

How can we investigate equal-area rectangles?

On square grid paper, draw rectangles whose side lengths are whole numbers, meaning zero and the positive counting numbers, and whose area is 24 square units. Compare their boundary lengths. The area condition controls the product, meaning the result of multiplying the side lengths; perimeter uses their sum.

The same investigation can be made with area 32 cm². Keep the required area fixed when comparing the perimeters. This is a useful way to check the mistaken idea that equal area must mean equal boundary length.

What do cutting, rejoining and tangrams teach us about area?

A tangram is a set of seven geometric pieces that can be rearranged to form shapes. Comparing these pieces by placing them over one another helps us recognise equal areas even when the outlines look different.

A square is divided into seven coloured pieces labelled A, B, C, D, E, F and G. The labels identify pieces rather than points. A and B are the two large triangles; C and E are the two small triangles.

Reference: NCERT Class 6, page 139

Pieces A and B have equal areas, and pieces C and E have equal areas. Pieces C and E together exactly cover piece D. Therefore, D has twice the area of either C or E.

What stays unchanged when pieces are rearranged?

If all seven pieces are rearranged without gaps or overlaps, the total area is the sum of the same seven piece areas. The outer shape can change while that total stays unchanged. Perimeter needs a separate check because different edges may become joined inside.

Another activity starts with a rectangular paper piece measuring 6 cm by 4 cm. It is cut into two equal rectangles, each measuring 6 cm by 2 cm. Joining the short ends makes the long arrangement shown in the activity, with perimeter 28 cm.

How should we count a rejoined boundary?

Follow the outside edge of the completed arrangement. A join between the paper pieces lies inside the new figure, so adding both original perimeters would also count edges that are no longer outside.

The activity asks for an arrangement with perimeter 22 cm. The important distinction is between preserving the paper and preserving the boundary. Rejoining the same pieces preserves their combined area, but it can produce a new perimeter.

How do house plans and area puzzles combine these ideas?

A plan shows the arrangement of spaces viewed from above. The house plans divide rectangular plots into rooms and other regions. Their measurements allow missing lengths to be recovered and areas to be compared.

The abbreviation ft means feet, a unit of length; “sq ft” means square feet, a unit of area. Use each plan's stated units consistently. A room's dimensions are its side measurements, while its area describes the region it occupies.

How can a room's missing side be found?

Charan's small bedroom has one side of 15 ft and area 180 sq ft. Its other side is 180 ÷ 15 = 12 ft. This is the same reverse-area calculation used for a garden's missing width.

Sharan's kitchen measures 18 ft by 10 ft and has area 180 sq ft. Equal room areas do not require identical side lengths. The complete rectangular outlines provide a further comparison of area and perimeter.

The plans enclose equal total areas, but Sharan's outline has the larger perimeter. When using an overall rectangle, distinguish its outer boundary from the dividing lines between rooms. Internal room divisions are not extra sides of the outer rectangle.

How do area maze puzzles use shared lengths?

An area maze is a diagram puzzle in which adjoining regions provide clues to missing lengths or areas. Look for rectangles that share a height or width. Use a known area and a known side to find another side, then transfer that measurement where the diagram justifies it.

For regions split into smaller rectangles, add the separate areas. For a region left after removing another, subtract the removed area from the whole. In either method, check that the selected parts account for exactly the region requested.

Glossary

• Perimeter — The total distance along the boundary of a closed figure when travelled once.
• Area — The measure of the region enclosed inside a closed figure.
• Polygon — A closed plane figure whose boundary is made entirely of line segments.
• Rectangle — A four-sided figure with four right angles and equal opposite sides.
• Square — A four-sided figure with four equal sides and four right angles.
• Regular polygon — A polygon with all sides equal in length and all angles equal.
• Equilateral triangle — A triangle with three equal sides and three equal angles.
• Diagonal — A line segment joining two corners of a polygon that are not adjacent.
• Square unit — The area of a square whose sides each measure one length unit.
• Estimate — An approximate value, such as an area obtained using boundary-square counting conventions.
• Tangram — A set of seven geometric pieces that can be rearranged into different shapes.
• Rate — An amount stated for each unit, such as rupees for each metre.

Common errors and misconceptions

• Misconception: Length plus breadth gives a rectangle's whole perimeter. Correct: This covers two sides. Double the sum to include the other two sides.
• Misconception: Perimeter and area measure the same thing. Correct: Perimeter measures boundary length; area measures the enclosed region, generally in square units.
• Misconception: A diagonal grid interval equals a straight grid interval. Correct: The diagonal is longer. Count the two kinds of intervals separately.
• Misconception: A smaller running track guarantees a shorter total run. Correct: Multiply each track's perimeter by the number of rounds before comparing distances.
• Misconception: Four equal flower beds require subtracting one bed's area. Correct: Find one bed's area, multiply by four, then subtract that total.
• Misconception: Equal areas require equal perimeters. Correct: The nine-square arrangements have equal areas but boundary lengths of 12 and 20 units.
• Misconception: Rejoining pieces preserves the sum of their separate perimeters as the outside boundary. Correct: Joined edges lie inside; trace the new outer boundary.
• Misconception: Every partly covered grid square counts as one. Correct: Ignore less than half, count more than half as one, and count exactly half as half.

Exam-style questions with model answers

Q1. A triangle has sides 4 cm, 5 cm and 7 cm. Find its perimeter and explain the operation used. [2 marks]

1. The perimeter is found by adding all three side lengths, because they together form the triangle's complete boundary.
2. Perimeter = 4 + 5 + 7 = 16 cm.

Q2. A wire forms a rectangle with length 5 cm and breadth 3 cm. It is straightened and bent into a square using the same whole wire. Find the square's side. [3 marks]

1. Find the total wire length from the original rectangle's perimeter: 2 × (5 + 3) = 16 cm. This includes both lengths and both breadths.
2. Bending the same whole wire changes its shape while retaining its length. Therefore, the square's perimeter is also 16 cm.
3. A square has four equal sides, so each side is 16 ÷ 4 = 4 cm.

Q3. Usha takes three complete rounds of a square park with side 75 m. Calculate her total distance, showing the distance for one round. [3 marks]

1. The boundary of the park has four equal sides. Its perimeter is therefore 4 × 75 = 300 m.
2. One complete round follows the entire boundary once, so the distance travelled in one round is 300 m.
3. Usha makes three complete rounds, so multiply the one-round distance by three: total distance = 3 × 300 = 900 m.

Q4. A rectangular floor is 5 m long and 4 m wide. A square carpet with side 3 m lies entirely on it. Find the floor area, carpet area and area not carpeted. [3 marks]

1. The floor is rectangular, so its area is length × width = 5 × 4 = 20 m².
2. The carpet is square, so its area is side × side = 3 × 3 = 9 m².
3. The carpet covers part of the floor. Subtract its area from the whole floor area: uncarpeted area = 20 − 9 = 11 m².

Q5. Four separate square flower beds, each of side 4 m, occupy the four corners of rectangular land 12 m long and 10 m wide. Find the area remaining after excluding all four beds. Explain each step. [5 marks]

1. The complete piece of land is a rectangle. Its area is length × width = 12 × 10 = 120 m².
2. Each flower bed is square. Its area is side × side = 4 × 4 = 16 m².
3. There are four beds with the same area. Their combined area is 4 × 16 = 64 m².
4. The beds occupy separate parts of the land, so subtract their combined area from the whole: 120 − 64 = 56 m².
5. The remaining area is therefore 56 m². This is the uncovered region, so the answer uses square metres rather than metres.

Q6. Akshi runs five rounds of a rectangular track measuring 70 m by 40 m. Toshi runs seven rounds of a rectangular track measuring 60 m by 30 m. Calculate each total distance and identify who runs farther. [5 marks]

1. Akshi's one-round distance is the perimeter of her rectangular track. It is 2 × (70 + 40) = 220 m.
2. She completes five rounds of that boundary, so her total distance is 5 × 220 = 1100 m.
3. Toshi's one-round distance is the perimeter of her own track. It is 2 × (60 + 30) = 180 m.
4. She completes seven rounds, so her total distance is 7 × 180 = 1260 m.
5. Toshi runs farther because 1260 m exceeds 1100 m. The smaller perimeter is outweighed by the greater number of rounds in this comparison.

Q7. A rectangular plot is 500 m long and 200 m wide. Tiling costs ₹8 per hundred square metres. Calculate the total tiling cost, showing how you use the rate. [4 marks]

1. The rectangular plot's area is length × width = 500 × 200 = 100,000 m².
2. The rate is for each hundred square metres, so divide the whole area into groups of that size.
3. Number of hundred-square-metre groups = 100,000 ÷ 100 = 1000.
4. Each group costs ₹8. Therefore, total tiling cost = 1000 × 8 = ₹8000, rather than charging ₹8 for every single square metre.

Q8. Two rectangular house outlines measure 35 ft by 30 ft and 42 ft by 25 ft. Calculate their areas and perimeters, then explain what their comparison shows. [5 marks]

1. The first rectangle's area is 35 × 30 = 1050 sq ft. This measures the region inside its outline.
2. The second rectangle's area is 42 × 25 = 1050 sq ft. Therefore, the two outlines enclose equal areas.
3. The first perimeter is 2 × (35 + 30) = 130 ft, which measures its complete outer boundary.
4. The second perimeter is 2 × (42 + 25) = 134 ft, so its outer boundary is longer.
5. The comparison shows that equal areas can have different perimeters. Both products are 1050, but the different sums of side lengths produce different boundary measurements.

Key takeaways

• Perimeter measures a complete boundary once; a polygon's perimeter is the sum of all its side lengths.
• A rectangle's perimeter is twice length plus breadth together; a square's perimeter is four times one side.
• Multiply the perimeter by the number of complete rounds to find the total distance travelled.
• Area measures an enclosed region and is generally expressed in square units; rectangle area is length times width.
• Subtract the combined area of covered or removed parts from the whole to find a remaining area.
• Grid estimates distinguish complete squares, portions smaller than half, portions larger than half, and exact halves.
• A rectangle's diagonal divides it into two triangles of equal area, each half of the rectangle.
• Rearranging the same pieces can preserve total area while changing the outside boundary and therefore the perimeter.

Test yourself

What is the perimeter of a square photo frame with side 1 m?

Its four equal sides give a perimeter of 4 × 1 = 4 m.

A rectangle has perimeter 14 cm and breadth 2 cm. What is its length?

Half the perimeter is 7 cm. Subtract the breadth: length = 7 − 2 = 5 cm.

A triangle has perimeter 55 cm and two sides of 20 cm and 14 cm. Find the third side.

Subtract the two known lengths from the total: third side = 55 − 20 − 14 = 21 cm.

A 36 cm string forms an equal-sided hexagon. What is each side's length?

A hexagon has six sides, so each equal side measures 36 ÷ 6 = 6 cm.

A rectangular garden has length 25 m and area 300 m². Find its width.

Divide area by the known length: width = 300 ÷ 25 = 12 m.

How do you count a grid square with exactly half its area inside a shape?

Count that portion as half a square unit when estimating the enclosed area.

Why do straight and diagonal intervals on a square grid need separate counts?

A diagonal interval is longer than a straight interval, so the two kinds are not equal length units.

Can two figures have the same area but different perimeters?

Yes. The nine-unit-square arrangements both have area nine square units, but perimeters of 12 and 20 units.