Notes for SSC CGL examinations — measure every boundary, surface, sector, path and shaded piece with speed and precision.
Perimeter is the total length around a closed figure. It answers “How much fencing, ribbon or boundary?”
Area is the surface enclosed. It answers “How much floor, sheet, paint or land?”
Linear conversion is applied once; area conversion is applied twice.
| Figure | Perimeter / Circumference | Area | Extra high-value relation |
|---|---|---|---|
| Triangle, sides \(a,b,c\) | \(P=a+b+c\) | \(A=\frac12 bh\) | \(s=\frac{a+b+c}{2}\), \(A=\sqrt{s(s-a)(s-b)(s-c)}\) |
| Equilateral triangle | \(P=3a\) | \(A=\frac{\sqrt3}{4}a^2\) | \(h=\frac{\sqrt3}{2}a\) |
| Rectangle | \(P=2(l+b)\) | \(A=lb\) | \(d=\sqrt{l^2+b^2}\) |
| Square | \(P=4a\) | \(A=a^2=\frac{d^2}{2}\) | \(d=a\sqrt2\) |
| Parallelogram | \(P=2(a+b)\) | \(A=bh\) | \(d_1^2+d_2^2=2(a^2+b^2)\) |
| Rhombus | \(P=4a\) | \(A=bh=\frac12d_1d_2\) | \(a=\frac12\sqrt{d_1^2+d_2^2}\) |
| Kite, equal pairs \(a,a,b,b\) | \(P=2(a+b)\) | \(A=\frac12d_1d_2\) | Diagonals are perpendicular |
| Trapezium, parallel sides \(a,b\) | sum of four sides | \(A=\frac12(a+b)h\) | midline \(m=\frac{a+b}{2}\), so \(A=mh\) |
| Regular \(n\)-gon, side \(a\) | \(P=na\) | \(A=\frac12P\rho=\frac{na^2}{4\tan(\pi/n)}\) | \(\rho\) is apothem; central angle \(=\frac{360^\circ}{n}\) |
| Circle | \(C=2\pi r=\pi d\) | \(A=\pi r^2\) | \(d=2r\) |
| Sector of angle \(\theta\) | arc \(L=\frac{\theta}{360^\circ}2\pi r\) | \(A=\frac{\theta}{360^\circ}\pi r^2\) | sector perimeter \(=L+2r\) |
| Annulus | outer + inner boundary \(=2\pi(R+r)\) | \(A=\pi(R^2-r^2)\) | factor: \(\pi(R-r)(R+r)\) |
The height must meet the chosen base at a right angle. In an obtuse triangle, that perpendicular can fall outside the figure.
Use it when all three sides are known. First confirm the triangle inequality: the sum of any two sides must exceed the third.
Example: for sides \(13,14,15\), \(s=21\), so
In a right triangle, the two perpendicular legs \(p,q\) are automatically base and height. If hypotenuse \(c\) is needed, use \(c^2=p^2+q^2\).
Diagonals of both figures bisect each other. Rectangle diagonals are equal; square diagonals are equal, perpendicular and angle-bisecting.
For fixed perimeter, a square has the largest area among rectangles. For fixed area, a square has the smallest perimeter among rectangles.
A parallelogram’s area is base × perpendicular height, never usually side × side. A rhombus is a parallelogram with all sides equal.
For the same base and between the same parallels, a triangle has half the area of a parallelogram.
A trapezium has one pair of parallel sides. Its area equals average of parallel sides × height. A kite has two adjacent equal-side pairs.
Join the centre to all vertices. A regular \(n\)-gon becomes \(n\) congruent isosceles triangles. The perpendicular from the centre to a side is the apothem \(\rho\).
Special regular hexagon: six equilateral triangles, hence \(A=\frac{3\sqrt3}{2}a^2\).
Use the value of \(\pi\) directed by the question. When no direction is given, \(\frac{22}{7}\) is convenient for multiples of \(7\); otherwise \(3.14\) or symbolic \(\pi\) may simplify better.
Diameter doubles the radius, but area becomes four times because radius is squared.
A sector is the same fraction of a circle as its central angle is of \(360^\circ\).
Sector perimeter includes arc + two radii.
A circular segment is bounded by a chord and arc:
An annulus is the region between concentric circles. Subtract the small disk from the large disk.
If uniform path width is \(x\), then \(R-r=x\). Boundary length of the material includes both circular edges.
Area: internal dividing lines do not matter. Perimeter: count only exposed boundary unless an inner hole is explicitly part of the boundary.
For an inside path, require \(l\gt2x\) and \(b\gt2x\). A path around a circular field is handled as an annulus.
If every linear dimension is multiplied by \(k\), all similar lengths and perimeters multiply by \(k\), while areas multiply by \(k^2\).
Thus area ratio \(m:n\) gives corresponding length ratio \(\sqrt m:\sqrt n\).
If rectangle length changes by \(x\%\) and breadth by \(y\%\), use signed values: decrease is negative.
Example: length rises \(20\%\), breadth falls \(10\%\): area change \(=20-10-2=8\%\) increase.
Fencing uses perimeter rate; flooring and painting use area rate. Convert units before multiplying.
If perimeter rises by \(20\%\), the scale factor is \(1.2\), so area rises by \(1.2^2-1=44\%\). No original dimension is needed.
If radii are \(14\) and \(7\), avoid squaring: \(\pi(7)(21)=147\pi=462\) using \(\pi=22/7\).
Arc and sector share the same fraction \(\theta/360^\circ\). Eliminate the angle:
This works directly when radius and arc length are given.
Cancel the common factor \(\frac12\).
Same height → compare bases only. Same base → compare heights only. Similar triangles with side ratio \(p:q\) have area ratio \(p^2:q^2\).
For a thin path of width \(x\), area is approximately perimeter × width. Exact rectangle formulas add or subtract corner squares:
Here \(P=2(l+b)\) is the original rectangle perimeter.
If diagonal is \(10\), area is immediately \(100/2=50\). If a square is inscribed in a circle, its diagonal equals the circle’s diameter.
Use signs. For equal rise and fall \(x\%\), net area change is \(-x^2/100\%\), always a decrease.
Example: \(+20\%,-20\%\Rightarrow-4\%\).
Write units beside every value. Convert the rate or the measurement—never both.
If tiles cover \(0.25\,\mathrm{m}^2\) each, number of tiles \(=\text{floor area}/0.25=4\times\text{floor area}\).
“A circular park is \(28\,\mathrm{m}\) across” gives the diameter, so \(r=14\), not \(28\).
Half circumference is only the curved part. A closed semicircle perimeter includes its diameter.
The arc alone is not the sector perimeter. Add both radii. Conversely, do not add radii when only “length of arc” is asked.
In triangle, parallelogram and trapezium area formulas, \(h\) means perpendicular distance. A sloping side cannot replace it unless perpendicular.
When pieces are joined, their common edge disappears from the outer perimeter. Add only the exposed boundary.
Because \(1\,\mathrm{m}=100\,\mathrm{cm}\), candidates often choose \(100\,\mathrm{cm}^2\). The correct area factor is squared.
Outside path increases both dimensions by \(2x\); inside path decreases both by \(2x\). Using \(x\) instead of \(2x\) misses one side.
A \(20\%\) rise followed by \(20\%\) fall does not restore area. Products, not sums, decide.
Sides \(2,3,5\) do not form a proper triangle because \(2+3\) is not greater than \(5\). Heron’s radicand becomes zero.
Perimeter patrols the edge like a fence. Area fills the inside like paint.
Heron carries four factors under one root: \(s\), then \(s-a\), \(s-b\), \(s-c\).
Visualize the parallel sides of a trapezium as railway rails. Average their lengths, then multiply by the separation.
When perpendicular diagonals cross, multiply them and cut the result in half.
The outer ring is linear: \(2\pi r\). The whole pizza is surface: \(\pi r^2\).
Every sector takes \(\theta\) out of \(360\) shares—from circumference for arc, from circle area for sector.
Stretching every length by \(k\) stretches perimeter once, but area in two directions—hence \(k^2\).
For uniform width \(x\), outside dimensions become \(+2x\); inside dimensions become \(-2x\).