Geometry is the study of points, lines, surfaces, and solids. For aptitude tests it splits into Pure Geometry (angles, properties, theorems) and Mensuration (area, perimeter, volume).
| Concept | Definition |
|---|---|
| Parallel Lines | Two lines in a plane that never intersect, no matter how far extended |
| Perpendicular Lines | Two lines that intersect at 90° |
| Sum of Angles in a Triangle | Interior angles always add to 180° |
| Supplementary Angles | Two angles whose sum is 180° |
| Exterior Angle of a Regular Polygon | Each exterior angle = 360° / n (n = number of sides) |
| Type | Property |
|---|---|
| Equilateral | All sides equal, all angles 60° |
| Isosceles | Two sides equal, angles opposite them equal |
| Right-Angled | One angle is 90° — follows Pythagoras theorem |
For a right-angled triangle with base b, height h, hypotenuse
c:
c2 = b2 + h2
∠BIC = 90° + (∠A / 2)
If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those sides in the same ratio.
If DE || BC, then AD/DB = AE/EC.
| Shape | Key Property | Area Formula |
|---|---|---|
| Rectangle | Opposite sides equal, all angles 90° | L × W |
| Square | All sides equal, all angles 90°, diagonals equal |
Side2 or
(Diagonal2)/2
|
| Parallelogram | Opposite sides parallel | Base × Height |
| Rhombus | All sides equal, diagonals bisect at 90° |
(1/2) × d1 × d2
|
| Trapezium | One pair of parallel sides (a, b) | (1/2) × (a + b) × Height |
A quadrilateral whose vertices all lie on a circle. The sum of opposite angles is 180° (supplementary).
6 equal sides — consists of 6 equilateral triangles. Area =
[ (3√3) / 2 ] × Side2
2πr (r = radius)πr2
2 × r
A sector is a slice of the circle defined by a central angle
θ.
Area of sector: (θ / 360) × πr2
t (PQ) touches at P and
a line through center O cuts the circle, use right-triangle
OQ2 = OP2 + PQ2.
For two circles with radii r1, r2 separated by distance d:
Length of direct common tangent:
√[ d2 − (r1 − r2)2 ]
(x1, y1) and
(x2, y2):√[ (x2 − x1)2 + (y2 − y1)2
]
Key concept (recasting): when one shape is melted to form another (e.g., sphere to cone), the volume remains constant.
| Solid | Volume | Surface Area / Notes |
|---|---|---|
| Cube (side a) |
a3
|
Total SA = 6a2, Vertices = 8
|
| Cylinder (r, h) | πr2h |
Curved SA = 2πrh |
| Cone (r, h) | (1/3)πr2h |
Slant l = √(h2 + r2), Curved SA = πrl
|
| Sphere (r) |
(4/3)πr3
|
Surface Area = 4πr2
|
| Tetrahedron (a) |
a3 / (6√2)
|
Regular triangular pyramid |
A cone with the top sliced off parallel to the base. With bottom radius R, top
radius r, slant height l:
Total SA: π(R + r)l + πR2 + πr2
If a path of width w is built outside a rectangle (L, B):
(L + 2w) and
(B + 2w)
(Outer Area) − (Inner Area)| Symbol | Meaning |
|---|---|
l, b, h |
Length, breadth, height |
a, s |
Side length |
r, R |
Radius |
d |
Diagonal or distance |
θ |
Angle |
π |
Pi (≈ 22/7 or 3.14) |
Each exterior angle of a regular polygon with n sides is?
In any triangle, ∠BIC where I is the incenter equals?
If DE || BC in ΔABC, Thales says?
Area of a rhombus with diagonals d1, d2 is?
Area of a regular hexagon with side s is?
Area of a sector with central angle θ and radius r is?
Length of direct common tangent for two circles (r1, r2, distance d) is?
A point moving equidistant from a fixed point and a fixed line traces a?
When a solid is melted and recast into another shape, what remains constant?
Pure geometry ⇒ angles/properties ; Mensuration ⇒ area/perimeter/volume
Parallel ⇒ never meet ; Perpendicular ⇒ 90°
Triangle interior ⇒ 180° ; Supplementary ⇒ sum 180° ; Regular n-gon exterior ⇒ 360°/n
a) Equilateral ⇒ all sides/60° ; Isosceles ⇒ 2 sides equal ; Right ⇒ 90° → Pythagoras
Pythagoras ⇒ c² = b² + h² (c = hypotenuse)
b) Incenter ⇒ angle bisectors → incircle ; ∠BIC = 90° + A/2
Circumcenter ⇒ perp bisectors → circumcircle
c) Thales ⇒ DE || BC → AD/DB = AE/EC
Rectangle ⇒ L×W ; Square ⇒ s² or diag²/2 ; Parallelogram ⇒ base×height
Rhombus ⇒ (1/2)d1d2 ; sides equal, diags bisect 90°
Trapezium ⇒ (1/2)(a+b)×h (a,b = parallel sides)
Cyclic ⇒ vertices on circle ; opposite angles sum 180°
Hexagon ⇒ 6 equilateral triangles ; area = (3√3/2)s²
Circumference ⇒ 2πr ; Area ⇒ πr² ; Diameter ⇒ 2r
Sector (angle θ) ⇒ (θ/360)πr²
Chord ⇒ perp from center bisects chord
Tangent ⇒ ⊥ radius at contact ; OQ² = OP² + PQ²
Direct common tangent ⇒ √[ d² − (r1−r2)² ]
Distance ⇒ √[ (x2−x1)² + (y2−y1)² ]
Locus ⇒ path under condition ; (eg equidistant point & line → parabola)
Recasting ⇒ volume constant
Cube ⇒ a³ ; SA 6a² ; vertices 8
Cylinder ⇒ πr²h ; curved 2πrh
Cone ⇒ (1/3)πr²h ; l = √(h²+r²) ; curved πrl
Sphere ⇒ (4/3)πr³ ; SA 4πr²
Tetrahedron ⇒ a³/(6√2)
Frustum ⇒ π(R+r)l + πR² + πr²
Square in circle ⇒ diagonal = diameter
Sphere in cube ⇒ diameter = side
Path outside rect (L,B) width w ⇒ outer (L+2w)(B+2w) ; path area = outer − inner
l,b,h ⇒ length/breadth/height ; a,s ⇒ side
r,R ⇒ radius ; d ⇒ diagonal/distance ; θ ⇒ angle ; π ≈ 22/7
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