Big picture
Fencing a garden and laying grass inside it involve different measurements. Perimeter measures boundary length, while area measures the surface enclosed. A long narrow garden can have the same area as a compact one but require more fencing.
Choosing a formula begins with understanding the region, not merely spotting a familiar number. Draw the shape, mark relevant lengths and distinguish perpendicular height from a slant side. For composite figures, decide which boundaries are external and which regions should be included or removed.
Learning goals
Distinguish perimeter from area and use appropriate units.
Calculate areas of rectangles, parallelograms and triangles.
Calculate circumference and area of circles.
Solve composite-region and path problems.
Interpret cost calculations and reverse-measurement questions.
Core ideas
1. Units and meaning
Perimeter is the distance around a closed figure and uses length units such as cm or m. Area counts square units covering a surface and uses cm² or m². A 1 m square has sides 100 cm long, so its area is 100 × 100 = 10,000 cm². Therefore 1 m² = 10,000 cm², not 100 cm².
For a rectangular card of length 8 cm and breadth 3 cm, perimeter = 2(8 + 3) = 22 cm, while area = 8 × 3 = 24 cm². These numbers answer different questions and cannot be compared as though they have the same unit. Draw a boundary line for perimeter and shade the inside for area to reinforce the distinction.
2. Squares and rectangles
A rectangle has P = 2(l + b) and A = lb. A square of side s has P = 4s and A = s². A 12 m by 7 m classroom has floor area 84 m² and perimeter 38 m.
If a square has perimeter 36 cm, its side is 36/4 = 9 cm, giving area 81 cm². Do not square the perimeter. If a rectangle's area is 96 cm² and length 12 cm, breadth = 96/12 = 8 cm. Reverse problems are easiest when the equation is written before substitution. The area alone does not determine a unique rectangle unless another condition is given.
3. Parallelograms and perpendicular height
The area of a parallelogram is base × perpendicular height. A triangular piece cut from one end can be shifted to the other to form a rectangle, explaining the formula. The height is measured at right angles to the selected base, not necessarily along a visible side.
For base 15 cm and perpendicular height 6 cm, area = 15 × 6 = 90 cm². If its slant side is 8 cm, that side affects perimeter but not this base–height area calculation. Perimeter is 2(15 + 8) = 46 cm. Different choices of base require matching perpendicular heights; the final area remains unchanged when both pairs are correct.
4. Triangles
Two congruent triangles can form a parallelogram, giving triangle area = ½ × base × height. The perpendicular height may lie inside, on a side or outside the triangle depending on its shape and chosen base.
A triangle with base 14 cm and height 9 cm has area ½ × 14 × 9 = 63 cm². If area is 48 cm² and base 12 cm, 48 = ½ × 12 × h, so h = 8 cm. A right triangle with perpendicular sides 6 cm and 8 cm has area 24 cm². Using two arbitrary sides in place of base and perpendicular height is invalid unless their right-angle relationship is established.
5. Circle measurements
A circle consists of points at fixed distance r from its centre. That distance is the radius, and diameter d = 2r. The boundary length is circumference C = 2πr = πd. Its area is A = πr². Use the value of π specified in the question and state any approximation.
For r = 7 cm with π = 22/7, C = 2 × 22/7 × 7 = 44 cm and A = 22/7 × 49 = 154 cm². If diameter is 14 cm, first take radius 7 cm before squaring. Doubling radius doubles circumference but quadruples area. This scaling difference is useful for checking whether a result is reasonable.
6. Semicircles and boundary counting
A semicircle has half the circle's area, ½πr². Its curved arc length is πr, but its full perimeter includes the diameter: P = πr + 2r. For r = 7 cm, area is 77 cm², arc length 22 cm and full perimeter 36 cm using 22/7.
A common error is reporting half the circle circumference as the entire semicircle perimeter. Trace the closed boundary with a pencil: the straight diameter must be included unless the question specifically asks only for the curved part. In composite shapes, a shared diameter may become an internal line and should then be excluded from the external perimeter.
7. Composite regions
Split a complicated region into non-overlapping familiar shapes, or subtract an unwanted region from a larger one. A 10 m by 8 m rectangle with a 3 m by 2 m corner removed has area 80 − 6 = 74 m². Area subtraction is valid because the smaller region is entirely within the larger.
Perimeter needs separate reasoning. Removing a corner rectangle may replace outer segments by equal new inner boundary segments, leaving perimeter unchanged in this particular axis-aligned arrangement. Never assume that subtracting area requires subtracting perimeter. Label each exposed edge and add it directly. An internal dividing line used for calculation is not part of the outer boundary.
8. Paths and borders
For a path of uniform width w outside a rectangle l by b, outer dimensions are l + 2w and b + 2w. Path area = outer rectangle area − inner rectangle area. Around a 20 m by 12 m lawn with a 1 m exterior path, outer area = 22 × 14 = 308 m², lawn area = 240 m², and path area = 68 m².
For an interior path, dimensions of the remaining central region decrease by 2w. A ring-shaped path between circles of radii 5 m and 3 m has area π(25 − 9) = 16π m², or 50.24 m² using π = 3.14. Width is a radial difference, so do not subtract it from a diameter without checking the diagram.
9. Costs and practical interpretation
Fencing cost usually uses boundary length; flooring or painting cost usually uses area. A 12 m by 7 m floor has area 84 m². At ₹150 per m², covering cost is 84 × 150 = ₹12,600. If fencing a garden of those dimensions at ₹80 per metre, cost is 38 × 80 = ₹3,040.
A gate reduces fencing length if the question says that opening is left unfenced. A 2 m gate in a 38 m boundary leaves 36 m of fencing, costing ₹2,880 at ₹80 per metre. Real projects may include waste, labour and thickness, but do not add unspecified quantities to a school problem. State what the quoted rate covers.
10. Scaling and investigation
Two rectangles 10 cm by 2 cm and 5 cm by 4 cm each have area 20 cm², but perimeters are 24 cm and 18 cm. Equal area does not imply equal perimeter. Draw both on squared paper and count boundary units as an activity.
If every linear dimension of a shape is multiplied by 3, perimeter becomes 3 times and area 9 times as large. This follows because area formulas multiply two lengths. For a paper-cutting activity, rearranging pieces without overlap or gaps preserves total area even when the outer perimeter changes. Such observations explain formulas and provide checks beyond memorisation.
A reliable approach to questions
Decide whether the task concerns boundary, surface or cost.
Draw and label the figure, converting all lengths to one unit.
Match bases with perpendicular heights and distinguish radius from diameter.
Add or subtract areas without double-counting overlaps.
Trace external edges separately for perimeter and attach the correct units.
Connections that matter
Fractions explains half-area relationships in triangles and semicircles.
Algebra finds unknown dimensions from area or perimeter equations.
Comparing quantities converts measurements into rates and practical costs.
Common mistakes
Reporting area in cm instead of cm² confuses dimensions.
Using a slant side as perpendicular height can overstate area.
Squaring a diameter in πr² makes the area four times too large.
Omitting a semicircle's diameter gives only arc length, not full perimeter.
Revision checklist
I can calculate and distinguish perimeter and area.
I can justify triangle and parallelogram formulas.
I can use circle and semicircle formulas correctly.
I can solve composite and uniform-path problems.
I can calculate costs and check scaling relationships.