Geometry and Mensuration

This topic covers shapes, angles, and measurement — a heavily tested area involving formulas for area, perimeter, volume, and the Pythagorean theorem.

• Angles:
• The angles of a triangle always sum to 180°.
• Complementary angles add up to 90°; supplementary angles add up to 180°.
• Angles on a straight line sum to 180°; angles around a point sum to 360°.
• Triangles: Classified by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). A right triangle has one 90° angle.
• Area formulas (space enclosed by a 2D shape):
• Triangle = 1/2 x base x height
• Rectangle = length x breadth
• Square = side^2
• Circle = π r^2 (where r = radius)
• Perimeter: The total distance around the boundary of a shape — sum of all sides for polygons; for a circle, this is the circumference = 2πr.
• Volume formulas (space occupied by a 3D shape):
• Cuboid = length x breadth x height
• Cylinder = π r^2 h
• Cube = side^3
• Pythagoras' theorem:
• In a right-angled triangle, if a and b are the two shorter sides (legs) and c is the hypotenuse (longest side, opposite the right angle), then a^2 + b^2 = c^2.
• Example: if a=3 and b=4, then c^2 = 9+16=25, so c=5 (the classic 3-4-5 triangle).
• Changes in dimensions (scaling): If a shape's linear dimensions are scaled by a factor k:
• Area scales by k^2
• Volume scales by k^3
• Example: doubling the side of a cube (k=2) increases its volume by 2^3 = 8 times.

Test tip: Always check whether a question asks for area, perimeter, or volume — mixing these up is a very common mistake under time pressure.