Rapid Revision · Quantitative Aptitude
Areas, perimeters, surface areas and volumes of the standard shapes. Memorize the sheet, watch your units, and these turn into free marks.
If you read nothing else tonight, read these 6 lines.
Every formula for mensuration in one place, each labelled so you know exactly when to reach for it. Screenshot it the night before.
Rectangle / square
rect: area = l x b, perimeter = 2(l+b) square: area = a^2
Triangle
area = 1/2 x base x height equilateral = (sqrt3 / 4) x a^2
Circle
area = pi x r^2 circumference = 2 x pi x r
Cuboid / cube
cuboid: V = lbh, TSA = 2(lb+bh+hl) cube: V = a^3, TSA = 6a^2
Cylinder
V = pi r^2 h CSA = 2 pi r h TSA = 2 pi r (r + h)
Cone
V = 1/3 pi r^2 h CSA = pi r l where l = sqrt(r^2 + h^2)
Sphere
V = 4/3 pi r^3 surface area = 4 pi r^2
6 cards, each one idea: what it is, a worked example, and the trap to dodge.
Rectangles, squares, triangles and circles cover almost every 2D question. Keep the equilateral-triangle and circle formulas at your fingertips.
Area of a circle with r = 7: pi x 49 = 22/7 x 49 = 154.
Trap: Perimeter grows linearly but area grows with the square: doubling a square's side quadruples its area.
For solids, keep volume (space inside) and surface area (skin outside) separate. Cuboid, cube, cylinder, cone and sphere are the exam set.
Volume of a cylinder r = 7, h = 10: pi x r^2 x h = 22/7 x 49 x 10 = 1540.
Trap: A cone's volume is one-third of the cylinder with the same base and height: V = 1/3 x pi x r^2 x h.
Diagonal of a rectangle = sqrt(l^2 + b^2). Space diagonal of a cuboid = sqrt(l^2 + b^2 + h^2). These unlock the trickier geometry questions.
Space diagonal of a cube of side a = a x sqrt3.
For a path of width w around or inside a rectangle, compute the big rectangle minus the small one. An outer path adds 2w to each dimension; an inner path subtracts it.
Path of width 2 around a 10 x 6 lawn: (14 x 10) - (10 x 6) = 140 - 60 = 80 sq units.
Trap: The width adds to BOTH ends of a side, so length grows by 2w, not w.
A pipe or ring is an outer solid minus an inner one. A toy or capsule is often a cone or hemisphere stacked on a cylinder, so add the parts up.
Volume of a pipe = pi x h x (R^2 - r^2), where R and r are the outer and inner radii.
If every length scales by k, area scales by k^2 and volume by k^3. This is the fastest route to any 'ratio of areas or volumes' question.
Two spheres with radii in ratio 1:2 have volumes in ratio 1:8.
A recap is not practice. These are the creators we rate for real depth on mensuration; full credit to each.
Each recap takes 3 minutes; the full set covers everything the first round tests. And when the test is cleared, your resume takes the next screen.
Original content by OptiResume; facts and formulas are common knowledge, the wording is ours. Go-deeper links go to creators we rate; we are not affiliated with them.