Perimeter, area and volume of standard shapes and solids, plus core angle, triangle, and circle results, with worked examples and practice
A formula-heavy topic that rewards a clean memorised table. Mensuration is measurement of length, area, and volume; geometry supplies the angle and triangle facts behind it. Most CAPF questions plug numbers into one of the formulas below, so the table is the topic. Use π as 22/7 or 3.14 as the question prefers, and build on the arithmetic in number system and simplification.
| Shape | Perimeter | Area |
|---|---|---|
| Square (side a) | 4a | a squared |
| Rectangle (l, b) | 2(l + b) | l times b |
| Triangle (sides a, b, c) | a + b + c | (1/2) times base times height |
| Equilateral triangle (side a) | 3a | (√(3)/4) times a squared |
| Parallelogram (base b, height h) | 2(sum of adjacent sides) | b times h |
| Rhombus (diagonals d1, d2) | 4 times side | (1/2) times d1 times d2 |
| Trapezium (parallel sides a, b, height h) | sum of all sides | (1/2) times (a + b) times h |
| Circle (radius r) | circumference = 2 times π times r | pi times r squared |
For a triangle with all three sides known, Heron's formula gives area = √( s(s minus a)(s minus b)(s minus c) ), where s = (a + b + c)/2 is the semi-perimeter.
| Solid | Curved or lateral surface area | Total surface area | Volume |
|---|---|---|---|
| Cube (side a) | 4 a squared (lateral) | 6 a squared | a cubed |
| Cuboid (l, b, h) | 2h(l + b) (lateral) | 2(lb + bh + hl) | l times b times h |
| Cylinder (radius r, height h) | 2 times pi times r times h | 2 times pi times r times (r + h) | pi times r squared times h |
| Cone (radius r, height h, slant l) | pi times r times l | pi times r times (r + l) | (1/3) times π times r squared times h |
| Sphere (radius r) | 4 times pi times r squared | 4 times pi times r squared | (4/3) times π times r cubed |
| Hemisphere (radius r) | 2 times pi times r squared (curved) | 3 times pi times r squared | (2/3) times π times r cubed |
For a cone, slant height l = √( r squared + h squared ). The diagonal of a cuboid is √( l squared + b squared + h squared ); the diagonal of a cube of side a is a times √(3).
| Result | Statement |
|---|---|
| Angles on a straight line | sum to 180° |
| Angles around a point | sum to 360° |
| Angle sum of a triangle | 180° |
| Angle sum of a quadrilateral | 360° |
| Angle sum of an n-sided polygon | (n minus 2) times 180° |
| Exterior angle of a triangle | equals the sum of the two opposite interior angles |
| Pythagoras theorem | in a right triangle, hypotenuse squared = sum of squares of the other two sides |
| Angle in a semicircle | 90° |
| Sum of exterior angles of any polygon | 360° |
Common Pythagorean triples to recognise on sight: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and their multiples such as (6, 8, 10).
A triangular plot has sides 13 m, 14 m, and 15 m. Find its area.
s = (13 + 14 + 15)/2 = 42/2 = 21. Area = √( 21 times (21 minus 13) times (21 minus 14) times (21 minus 15) ) = √( 21 times 8 times 7 times 6 ). 21 times 8 = 168; 168 times 7 = 1176; 1176 times 6 = 7056. √(7056) = 84 square metres.
A cylindrical drum has radius 7 cm and height 10 cm. Find its volume (π = 22/7).
Volume = π times r squared times h = (22/7) times 49 times 10 = 22 times 7 times 10 = 1540 cubic cm.
Find the surface area of a sphere of radius 7 cm (π = 22/7).
Surface area = 4 times π times r squared = 4 times (22/7) times 49 = 4 times 22 times 7 = 616 square cm.
A ladder 13 m long leans against a wall with its foot 5 m from the base. How high up the wall does it reach?
This is a (5, 12, 13) triple. Height = √(13 squared minus 5 squared) = √(169 minus 25) = √(144) = 12 m.
Find the sum of the interior angles of a regular hexagon, and each interior angle.
Sum = (6 minus 2) times 180 = 4 times 180 = 720°. Each angle = 720 / 6 = 120°.
A cone has base radius 3 cm and height 4 cm. Find its volume and slant height (π = 3.14).
Volume = (1/3) times 3.14 times 9 times 4 = (1/3) times 113.04 = 37.68 cubic cm. Slant l = √(3 squared + 4 squared) = √(9 + 16) = √(25) = 5 cm.