Mensuration | ICSE Class 7 Maths Notes

This note covers perimeter, area measured with unit squares, rectangles and squares, conversion of area units, parallelograms, triangles, circle circumference, circle area, circular rings and combined figures.

What do perimeter and area measure?

Definition: Perimeter is the distance around the boundary of a closed plane figure. Area measures the region enclosed by that boundary. A plane figure is a flat shape, and a closed figure has a boundary that encloses a region.

Mensuration concerns the measurement of shapes. In a question about putting lace around a tablecloth, the required measurement is perimeter. In a question about covering a floor with carpet, the required measurement is area. Identify the region or boundary before choosing a formula.

How is a perimeter calculated?

A polygon is a closed plane figure made of straight line segments. Add the lengths of all its sides to find its perimeter. A triangle has three sides, so a triangle with sides 4 centimetres, 5 centimetres and 7 centimetres has perimeter 16 centimetres.

The abbreviation cm means centimetre, and m means metre. Both are units of length. Perimeter is a length, so its unit is cm or m when the side measurements use those units. Area uses square units, which measure coverage instead.

A rectangle has four right angles and opposite sides equal. A right angle measures 90 degrees. A square has four equal sides and four right angles. Let P mean perimeter, l mean rectangle length, b mean rectangle breadth, and s mean square side length.

P = 2(l + b) for a rectangle, because each of its two different side lengths occurs twice. For a square, P = 4s. Symbols placed together indicate multiplication, and brackets show a calculation to complete first.

Worked example 1. A rectangular tablecloth is 3 m long and 2 m wide. Find the length of lace needed to go around it once.

Answer: Use the perimeter, P = 2(l + b). Substitute the two side lengths: P = 2 × (3 + 2) = 10 m. The lace must cover the complete boundary, so 10 m is required.

A regular polygon has equal sides and equal angles. Its perimeter is the number of sides multiplied by one side length. For other polygons, use the individual side lengths rather than assuming that all sides are equal.

How can unit squares measure an area?

A unit square is a square whose side measures one chosen unit of length. Its area is one square unit. If the side is 1 cm, its area is one square centimetre, written 1 cm². The superscript ² indicates a squared quantity.

A square grid is a pattern of equal squares covering a flat surface. Comparing the number of equal squares inside different shapes allows their areas to be compared. Use the same square size for both shapes so that the counts represent the same unit.

How are incomplete squares counted?

For a shape whose boundary does not follow the grid, some squares are only partly covered. The following counting convention gives an approximate area, meaning an estimate rather than a guaranteed exact measurement. Trace the outline onto transparent paper and place it over squared paper.

1. Count each complete small square inside the shape as one square unit.
2. Ignore a covered portion that occupies less than half a small square.
3. Count a covered portion greater than half a small square as one square unit.
4. Count a covered portion that is exactly half a small square as half a square unit.

Add the contributions to obtain the area. If partly covered squares are rounded up or ignored, report the result as approximate. If every covered square is whole or exactly half covered, counting them gives an exact area.

Why are formulas useful?

For a rectangle filled completely by rows of unit squares, a formula replaces repeated counting. Count how many squares fit along a row and how many rows fit across the breadth. Multiplying those counts gives the total number of unit squares in the rectangle.

The same idea explains why area measures the inside of the boundary. Counting just the squares that touch the boundary would leave out the interior. Conversely, adding the lengths of the boundary segments would measure perimeter rather than the covered region.

Note: A grid estimate and a formula calculation answer the same type of question, how much region is enclosed. Keep the distinction between an approximate count for an irregular outline and an exact formula applied to given dimensions.

How are the areas of rectangles and squares found?

Result: Rectangle and square area formulas

Let A mean area. Continue using l for rectangle length and b for rectangle breadth. The rectangle formula is A = l × b. For a square with side s, it becomes A = s², where s² means s multiplied by itself.

The rectangle formula counts unit squares arranged in rows. Each row contains the number of units in the length, and the number of rows is the number of units in the breadth. A square is a special rectangle with equal length and breadth.

Write both dimensions in the same length unit before multiplying. Multiplying metres by metres produces square metres, written m². Multiplying centimetres by centimetres produces square centimetres. The answer states the amount of surface covered, not a distance around its edge.

How can an uncovered area be calculated?

When one region lies completely inside another, subtract the inner area from the outer area to find what remains. Calculate the areas separately first. Keep their units the same so that the subtraction compares like quantities.

Worked example 2. A rectangular floor is 5 m long and 4 m wide. A square carpet of side 3 m is laid on it. Find the area of the floor that is not carpeted.

Answer: Floor area = 5 × 4 = 20 m². Carpet area = 3 × 3 = 9 m². Uncovered floor area = 20 − 9 = 11 m². The minus sign, −, means subtract.

The calculation separates the whole floor from the part covered by the carpet. Adding the two areas would count the carpeted part again. Check the subtraction by adding the uncovered area to the carpet area: their total is the whole floor area.

An unknown breadth can also be found from area and length. Rearranging the rectangle formula gives b = A ÷ l, where ÷ means divide. This reverses multiplication. The resulting breadth is a length and must therefore be reported in a length unit.

How are square units converted correctly?

A conversion expresses the same measurement in another unit. One metre equals 100 centimetres. However, one square metre is the area of a square measuring 100 cm in both directions. Its area is therefore 100 × 100 = 10,000 cm².

The length conversion acts in both directions of the area. This explains why the area conversion factor is squared. A square centimetre has sides of 10 millimetres, abbreviated mm, so it contains 100 square millimetres, written mm².

Which area conversions are useful?

A kilometre, abbreviated km, equals 1,000 m. A hectare is an area unit equal to 10,000 m². It measures land area rather than a side length. Keep the distinction between the kilometre and the square kilometre when reading a question.

Worked example 3. Express one hectare in square centimetres.

Answer: One hectare = 10,000 m², and each square metre = 10,000 cm². Therefore one hectare = 10,000 × 10,000 = 10,00,00,000 cm². Both conversion steps concern area, so the final unit remains a square unit.

How should mixed measurements be handled?

Choose one length unit before using an area formula. Convert each given length into that unit, then multiply. Alternatively, calculate an area in one consistent square unit and convert the completed area using the appropriate area conversion factor.

To move from a larger area unit to a smaller one, multiply by the number of smaller units it contains. To move in the reverse direction, divide by that number. A larger numerical count of smaller squares can represent exactly the same region.

Note: Do not apply a length conversion directly to an area. The statement 1 m = 100 cm does not mean that 1 m² = 100 cm². Think of the rows and columns of small squares.

How does base and height determine a parallelogram's area?

A parallelogram is a four-sided figure with both pairs of opposite sides parallel. Parallel lines stay the same distance apart and do not meet when extended. Any side may be chosen as the base, the side used as the reference for measuring height.

The height, also called the altitude, is the perpendicular distance between the base and the opposite parallel side. Perpendicular lines meet at a right angle. A sloping side is not generally this distance, so identify the right-angle height before substituting numbers.

Result: Area is base multiplied by perpendicular height

For this formula, let b mean the chosen base length and h mean its corresponding height. Then A = b × h. Here b represents a base, whereas in the rectangle formula it represented breadth. The meaning is stated with the shape being measured.

The horizontal lower side is labelled b. A dashed vertical double-headed arrow labelled h runs from the level of the upper side to the level of the lower side, distinguishing height from the sloping edge.

See Fig. 9.3 in your NCERT textbook

Can a different side be used as the base?

Yes, but it must be paired with its own perpendicular height. Changing the chosen base does not change the region of the parallelogram. Therefore the two base-and-height products give the same area even when the base lengths and corresponding heights differ.

Worked example 4. A parallelogram has adjacent sides 8 cm and 6 cm. Its perpendicular height corresponding to the 8 cm side is 4 cm. Find its area and the height corresponding to the 6 cm side.

Answer: Area = 8 × 4 = 32 cm². For the second base, 6 × h = 32. Therefore h = 32 ÷ 6 = 16/3 cm. The slash in 16/3 represents a fraction, sixteen divided by three.

Multiplying 8 cm by the sloping 6 cm side would not use the given perpendicular height. The correct calculation first establishes one area, then uses that same area with the other base. Check each height against the side to which it corresponds.

How is the area of a triangle calculated?

A triangle is a closed figure with three straight sides. Any side can be selected as its base. The corresponding height is the perpendicular distance from the opposite vertex, or corner, to the line containing that base.

Use the base length and the height belonging to that base. A line drawn from the opposite vertex to the base is not necessarily a height: it must meet the base line at a right angle. The height may meet an extension of the side.

Result: Triangle area is half the base-height product

With A as area, b as base length and h as the corresponding perpendicular height, A = ½ × b × h. The symbol ½ means one half. Multiplying the base and height without halving gives twice the required triangle area.

A triangle's area is half the area of a parallelogram having the same base and height. This comparison explains the factor ½. Keep both conditions: the same base alone is insufficient to establish that two triangles have equal areas.

How is an unknown height found?

When area and base are known, reverse the area formula. Multiply the area by two, then divide by the base. Thus h = 2A ÷ b. Similarly, if area and height are known, b = 2A ÷ h.

Worked example 5. A triangular piece of cardboard has area 90 cm² and a base of 20 cm. Find the altitude corresponding to that base.

Answer: Use h = 2A ÷ b. The height is (2 × 90) ÷ 20 = 9 cm. Check by substituting back: ½ × 20 × 9 = 90 cm², which matches the given area.

A right-angled triangle has one right angle. Its two sides meeting at that angle are perpendicular and can serve as a base-height pair. This is different from using any two sides of a general triangle as though they were perpendicular.

Perimeter still requires the lengths around the boundary. The base and height determine the area, but they do not by themselves provide every side length needed for the perimeter. Read carefully whether the question asks for distance around the triangle or its enclosed region.

What is the circumference of a circle?

A circle is a closed curve whose points are equally distant from a fixed point called its centre. Its radius is the distance from the centre to the circle. Its diameter is a straight segment across the circle through the centre.

Let r represent radius and d represent diameter. The diameter is twice the radius, so d = 2r. If the diameter is given, divide it by two to obtain the radius. Check which measurement is supplied before choosing a formula.

Result: Circumference depends on diameter or radius

The circumference is the distance around the circle. Let C represent circumference. The ratio of circumference to diameter is a constant, the same for every circle, represented by the Greek letter π, pronounced pi. A ratio here compares two lengths by dividing the first by the second.

The circumference formulas are C = πd and C = 2πr. They express the same relationship because d = 2r. Circumference is a length, so the answer uses the same length unit as the radius or diameter.

For numerical calculations, approximate values of π include 22/7 and 3.14. These are approximations, not exact equalities to π. Use the value requested in the problem consistently. A numerical answer obtained using it is an approximation to the circle's measurement.

Worked example 6. Find the circumference of a circle with diameter 5 cm, using π approximately equal to 3.14.

Answer: The diameter is already given, so use C = πd. With the stated approximation, C = 3.14 × 5 = 15.7 cm. This measures one complete journey around the circle.

How is a radius recovered from circumference?

Reverse C = 2πr to obtain r = C ÷ (2π). Divide by the whole product 2π. Dividing by π alone produces the diameter. Keep the unit of the resulting length and distinguish it from the area that may be calculated next.

Fencing around a circular garden uses circumference. Covering the circular region uses area. These tasks involve the same shape, but they require different formulas because one concerns the boundary and the other concerns the inside.

How is the area of a circle found?

The area of a circle means the area of the region enclosed by its circular boundary. With A as area and r as radius, the formula is A = πr². The expression r² means r × r, the square of the radius.

Use the radius even when the question gives a diameter. First calculate r = d ÷ 2, then multiply π by r × r. Squaring the diameter instead would use a different length and would not give the area required by this formula.

How do circumference and area calculations differ?

The radius occurs once in the circumference formula and twice as a factor in the area formula. The symbol ² does not mean multiply the radius by two. It means multiply the radius by itself. This distinction also explains the different units.

Worked example 7. A circle has circumference 33 cm. Find its area using π approximately equal to 22/7.

Answer: First find r = 33 ÷ (2 × 22/7) = 21/4 cm. Then A = (22/7) × (21/4) × (21/4) = 693/8 cm². This is the area calculated with the stated approximation for π.

The circumference is not a radius and cannot be substituted directly into the area formula. The first step changes the given information into the measurement needed by the second formula. Keep the fractional radius during the calculation instead of rounding it early.

How can the result be checked?

Check the radius by returning to the circumference formula. Multiplying it by 2π, using the same approximation, must reproduce the given circumference. Then check the area unit: because two radius lengths were multiplied, the result must be in square centimetres.

If a circle's radius doubles, its area becomes four times as large because both radius factors double. Its circumference becomes twice as large because there is only one radius factor. The two measurements therefore do not change by the same multiplier.

How is the area of a circular ring calculated?

A circular ring is the region between two circles with the same centre. Such circles are called concentric circles. The ring has an outer circular boundary and an inner circular boundary. A circular path around a pond provides this type of area problem.

Let R represent the outer radius and r represent the inner radius. The capital and lower-case letters distinguish the two distances from the common centre. Let w represent the uniform width of the ring, measured between its boundaries along a radius.

Result: Subtract the inner circle from the outer circle

The outer circle contains both the ring and the inner circular region. Therefore ring area is outer area minus inner area: A = πR² − πr². The ring width is w = R − r, so the inner radius can be found by subtracting width from outer radius.

Find both radii before calculating the areas. If the question gives a circumference, convert it to the appropriate radius first. The outer circumference corresponds to R; the inner circumference corresponds to r. Mixing them changes the dimensions of the ring.

Worked example 8. A circular pond is surrounded by a uniform circular path 2 m wide. The outer circumference of the path is 44 m. Find the inner circumference and the path area, using π approximately equal to 22/7.

Answer: Outer radius R = 44 ÷ (2 × 22/7) = 7 m. Inner radius r = 7 − 2 = 5 m. Inner circumference = 2 × (22/7) × 5 = 220/7 m. Path area = (22/7) × (7² − 5²) = 528/7 m².

Why is the order of subtraction important?

The outer region is larger and includes the part to be removed. Subtract the inner area from it to leave the path. The square of the width is not the difference of the two circle areas; width alone does not determine the ring's area.

Check the dimensions before checking arithmetic. The two radii should differ by the path width. Use length units for either circumference and square units for the ring area. The same approximation for π should be used throughout the linked calculations.

How are areas of combined figures calculated?

A combined figure is a region made from simpler shapes. Find its area by separating it into known regions without overlap, or by subtracting unwanted regions from a larger enclosing shape. Choose a method that uses the measurements actually supplied.

How should the parts be organised?

1. Identify the whole region and the part whose area is required.
2. Mark each known rectangle, square, triangle or circle and its dimensions.
3. Calculate the component areas using consistent square units.
4. Add the required non-overlapping areas or subtract the excluded areas, then check that every part is counted once.

For an outside rectangular path, the outer rectangle includes the inner rectangle and the path. A width added on opposite sides contributes twice to the corresponding outer dimension. Identify which strips increase the length and which increase the breadth.

A smaller rectangle lies inside a larger rectangle. The inner rectangle is labelled 30 m horizontally and 20 m vertically. The gap at the right is marked 8 m, and the lower gap is marked 5 m.

See Fig. 9.11 in your NCERT textbook

Worked example 9. A rectangular swimming pool measures 30 m by 20 m. Outside it, a path is 5 m wide along each 30 m side and 8 m wide along each 20 m side. Find the path area and the cost of cementing it at 200 rupees per m².

Answer: Outer length = 30 + 8 + 8 = 46 m. Outer breadth = 20 + 5 + 5 = 30 m. Outer area = 46 × 30 = 1,380 m². Pool area = 30 × 20 = 600 m². Path area = 780 m². Cost = 780 × 200 = 156,000 rupees.

How are openings excluded from an area?

A wall containing a door and a window can be treated as a whole rectangle with smaller rectangles removed. Subtract the area of each opening once. Their positions do not alter this area subtraction when the openings are separate and lie within the wall.

Worked example 10. A wall measures 5 m by 4 m. It contains a window measuring 1.5 m by 1 m and a separate door measuring 2.25 m by 1 m. Find the wall area to be painted, excluding both openings.

Answer: Whole wall area = 5 × 4 = 20 m². Window area = 1.5 × 1 = 1.5 m². Door area = 2.25 × 1 = 2.25 m². Paintable area = 20 − 1.5 − 2.25 = 16.25 m².

If two paths cross, adding their separate areas counts the crossing twice. Subtract that shared area once to obtain their combined area. In every addition or subtraction, ask which region each term represents rather than combining all the numbers in the question.

Glossary

• Perimeter — The total distance along the boundary of a closed plane figure, measured by going around it once.
• Area — The measure of the region enclosed by a closed plane figure, expressed in square units.
• Unit square — A square with side length one chosen unit, representing one square unit of area.
• Square grid — A pattern of equal squares used to count or estimate the area enclosed by a figure.
• Base — The chosen side of a triangle or parallelogram used with its corresponding height to calculate area.
• Altitude — The perpendicular height corresponding to a chosen base of a triangle or a parallelogram.
• Radius — The distance from the centre of a circle to any point on its circular boundary.
• Diameter — A straight segment joining two points on a circle through its centre, with length twice the radius.
• Circumference — The distance around a circle, equal to pi multiplied by its diameter.
• Pi — The constant ratio of a circle's circumference to its diameter, represented by the Greek letter π.
• Circular ring — The region between an outer circle and an inner circle sharing the same centre.
• Hectare — A unit used for land area that is equal to ten thousand square metres.

Common errors and misconceptions

• Misconception: Perimeter and area measure the same feature of a figure. Correct: Perimeter measures distance around the boundary; area measures the enclosed region. They require different units.
• Misconception: Equal perimeters guarantee equal areas. Correct: Shapes can have the same perimeter and different areas. Calculate the required measurement rather than inferring it from the other one.
• Misconception: A parallelogram's area is found by multiplying any two adjacent sides. Correct: Multiply a chosen base by its corresponding perpendicular height, which is not generally the sloping side.
• Misconception: A triangle's area equals base multiplied by height. Correct: Halve that product. Base multiplied by height is the area of a parallelogram with that base and height.
• Misconception: The diameter can be substituted for r in πr². Correct: The symbol r means radius. Divide a given diameter by two before using the circle-area formula.
• Misconception: Since 1 m equals 100 cm, 1 m² equals 100 cm². Correct: A square metre contains 100 rows of 100 square centimetres, giving 10,000 cm².
• Misconception: A circular path's area is found by squaring its width. Correct: Subtract the inner circle's area from the outer circle's area, using the two radii.
• Misconception: Counting partly covered grid squares always gives an exact area. Correct: Rounding covered portions to whole squares or ignoring them gives an approximate area. Counting only whole squares and exact half-squares gives an exact area.

Exam-style questions with model answers

Q1. Distinguish between perimeter and area, including the type of unit used for each. [2 marks]

1. Perimeter is the distance around the boundary of a closed figure. It is measured in length units such as metres.
2. Area measures the enclosed region. It is measured in square units such as square metres.

Q2. A rectangular tablecloth is 3 m long and 2 m wide. Calculate the length of lace needed to go around its boundary once, explaining your choice of measurement. [3 marks]

1. Lace is needed along the edge of the tablecloth, so the required measurement is the perimeter of the rectangle.
2. The perimeter formula is P = 2(l + b), where l is length and b is breadth. Substitute l = 3 m and b = 2 m.
3. Therefore P = 2 × (3 + 2) = 10 m. The required length of lace is 10 m.

Q3. A triangular piece of cardboard has area 90 cm² and base length 20 cm. Calculate the corresponding perpendicular height and check your answer. [3 marks]

1. For a triangle, A = ½ × b × h, where A is area, b is base and h is perpendicular height. Here 90 = ½ × 20 × h.
2. Rearrange to obtain h = (2 × 90) ÷ 20 = 9 cm. The height is a length, so it is expressed in centimetres.
3. Check the area: ½ × 20 × 9 = 90 cm². This agrees with the given cardboard area.

Q4. A parallelogram has adjacent sides 8 cm and 6 cm. The perpendicular height corresponding to the 8 cm side is 4 cm. Find its area and the height corresponding to the 6 cm side. [4 marks]

1. Use the base-height pair supplied: base 8 cm and perpendicular height 4 cm. The adjacent side of 6 cm is not the height for this calculation.
2. The area is base × corresponding height = 8 × 4 = 32 cm².
3. Let h be the height corresponding to the 6 cm side. The same parallelogram has the same area, so 6 × h = 32.
4. Dividing gives h = 32 ÷ 6 = 16/3 cm. This height is paired with the 6 cm base.

Q5. A circle has circumference 33 cm. Calculate its radius and area, using π approximately equal to 22/7. [4 marks]

1. Use C = 2πr, where C is circumference and r is radius. The given circumference therefore gives 33 = 2 × (22/7) × r.
2. Divide by 2 × (22/7): r = 33 × 7 ÷ 44 = 21/4 cm.
3. Use A = πr² for the area, substituting the radius rather than the circumference: A = (22/7) × (21/4) × (21/4).
4. Simplifying gives A = 693/8 cm². This numerical area uses the specified approximation for π and is reported in square centimetres.

Q6. A circular pond has a uniform circular path 2 m wide around it. The outer circumference of the path is 44 m. Find the inner circumference and the area of the path, using π approximately equal to 22/7. [5 marks]

1. Let R be the outer radius. Use the given outer circumference: 44 = 2 × (22/7) × R. Therefore R = 7 m.
2. The path width is the difference between the two radii. The inner radius, r, is therefore 7 − 2 = 5 m.
3. The inner circumference is 2πr = 2 × (22/7) × 5 = 220/7 m, using the stated approximation.
4. The area of the path is the outer circle's area minus the inner circle's area: πR² − πr².
5. Substitute both radii: path area = (22/7) × (49 − 25) = 528/7 m². The subtracted inner region is the pond.

Q7. A rectangular swimming pool is 30 m long and 20 m wide. An outside path is 5 m wide along each 30 m side and 8 m wide along each 20 m side. Find the area of the path and its cementing cost at 200 rupees per m². [6 marks]

1. The outer length includes an 8 m strip at each end of the pool: 30 + 8 + 8 = 46 m.
2. The outer breadth includes a 5 m strip on each side of the pool: 20 + 5 + 5 = 30 m.
3. The outer rectangle contains the pool and all the surrounding path. Its area is 46 × 30 = 1,380 m².
4. The pool's own area is 30 × 20 = 600 m². This region is excluded from the cementing calculation.
5. Subtract the inner region once: path area = 1,380 − 600 = 780 m². This leaves the complete surrounding path.
6. Multiply this area by the given cost per square metre: 780 × 200 = 156,000 rupees for cementing the path.

Q8. A rectangular wall measures 5 m by 4 m. It contains a window measuring 1.5 m by 1 m and a separate door measuring 2.25 m by 1 m. Calculate the area to be painted, excluding both openings. [5 marks]

1. Find the whole wall area using length multiplied by breadth: 5 × 4 = 20 m², before excluding either opening.
2. The window is a rectangle, so its area is 1.5 × 1 = 1.5 m². This area will not be painted.
3. The door is another rectangle, with area 2.25 × 1 = 2.25 m². It is separate from the window.
4. Add the two excluded areas: 1.5 + 2.25 = 3.75 m². Each opening is included once in this total.
5. The paintable area is whole wall area minus excluded area: 20 − 3.75 = 16.25 m². All dimensions were given in metres.

Key takeaways

• Perimeter measures the complete boundary of a closed figure; area measures the enclosed region and uses square units.
• Grid counting gives an approximate area when covered portions are rounded or ignored; counting only whole squares and exact half-squares gives an exact area.
• A rectangle's area is length multiplied by breadth; a square's area is its side length multiplied by itself.
• For a parallelogram or triangle, pair the chosen base with its corresponding perpendicular height before calculating the area.
• A circle's circumference is 2πr and its area is πr², where r is radius; use the stated approximation for π.
• Convert length units consistently before multiplying; converting a completed area requires the conversion factor for square units.
• A circular ring's area is the outer circle's area minus the inner circle's area, using the correct radius for each.
• For combined figures, add non-overlapping component areas or subtract excluded regions, making sure each part is counted once.

Test yourself

What measurement gives the length of fencing around a closed field?

The perimeter gives the distance around the field's boundary. For a circular field, this distance is called the circumference.

How is a grid square counted when exactly half of it lies inside a shape?

Count the covered portion as half a square unit when estimating the area using the stated grid convention.

Why does one square metre equal 10,000 square centimetres?

A square metre measures 100 cm in each direction. It contains 100 × 100 = 10,000 squares of area 1 cm².

Which height belongs in a parallelogram's area formula?

Use the perpendicular distance between the chosen base and the opposite parallel side, rather than assuming that a sloping side is the height.

What additional condition is needed for triangles with equal bases to have equal areas?

Their corresponding perpendicular heights must also be equal, since triangle area depends on both base and height.

What does π represent, and is 3.14 its exact value?

Pi is the constant ratio of a circle's circumference to its diameter. The value 3.14 is an approximation.

How do you calculate a uniform circular path's inner radius from its outer radius and width?

Subtract the path width from the outer radius. Both measurements must be expressed in the same length unit.

Why is the shared area subtracted when adding the areas of two crossing paths?

Adding the separate path areas counts the crossing twice. Subtracting the shared area once leaves it counted once in the combined area.