Mensuration & Basic Geometry
Mensuration & Basic Geometry is a frequently tested area in CUET General Test. Work through these free NTA-style sample questions with full answers and explanations, then attempt all 35 in a timed practice test to build exam-day speed.
Snapshot
- Mensuration and basic geometry ask for the perimeter and area of flat shapes, and the surface area and volume of solids, plus a handful of angle and triangle facts.
- Almost every mark is a formula recalled correctly and used with the right units: area is in square units, volume in cubic units, and the most common slip of all is using the diameter where the radius belongs.
- This page explains where the formulas come from so you can rebuild them, gives a LaTeX formula sheet, shortcuts and their limits, 15 worked examples, a traps table, 17 rapid-fire questions with answers and a one-week routine.
- Exam reality: +5 for a correct answer, −1 for a wrong one. Use π = 22/7 when the radius is a multiple of 7.
Part 1 — The core idea: measure by counting unit squares and unit cubes
Written and reviewed by Devendra Dubey, Mathematics & Commerce faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching). Prepared for the CUET UG General Test.
Where the formulas come from
Perimeter is a length: walk around the edge and add up the sides. Area is the number of unit squares that cover the surface, so it is a length multiplied by a length. Volume is the number of unit cubes that fill a solid, so it is a length multiplied three times. That is why the units are cm, cm² and cm³, and why doubling every length multiplies the area by 4 and the volume by 8.
A rectangle with sides l and b is covered by l rows of b squares, so the area is l × b. A triangle is half of a rectangle or parallelogram with the same base and height, so the area is ½ × b × h. A parallelogram can be cut and re-attached as a rectangle with the same base and height, so its area is b × h. A trapezium is half a parallelogram (two copies fit together to make one with base a + b), so the area is ½(a + b)h. A circle can be cut into thin sectors and rearranged into a near-rectangle of length πr and breadth r, which gives πr².
Why solids follow the same logic
A prism or cylinder is a stack of identical layers, so its volume is base area × height: πr²h for a cylinder. A cone fills exactly one third of the cylinder with the same base and height, which is why its volume is ⅓πr²h. A sphere of radius r fits in a cylinder of radius r and height 2r and takes two thirds of it, so its volume is ⅔ × πr² × 2r = (4/3)πr³.
Why the cone has a slant height
The curved surface of a cone unrolls into a sector whose radius is the slant height l, not the vertical height h. The slant height, the radius and the height form a right triangle, so $l = \sqrt{r^2 + h^2}$. This is why the Pythagorean triple 3-4-5 shows up so often in cone questions.
Pause and predict: the side of a square is increased by 10%. By what percent does its area increase: 10%, 20% or 21%?
Check your answer: 21%. Area is side squared, so the factor is $1.1 \times 1.1 = 1.21$. The same successive-change logic applies as in percentages: $10 + 10 + \dfrac{10 \times 10}{100} = 21$.
Part 2 — Formula sheet
Two-dimensional figures:
| Figure | Area | Perimeter |
|---|---|---|
| Square, side $a$ | $a^2$ | $4a$ |
| Rectangle, $l$ and $b$ | $lb$ | $2(l + b)$ |
| Triangle, base and height | $\dfrac{1}{2}bh$ | $a + b + c$ |
| Triangle, three sides (Heron) | $\sqrt{s(s-a)(s-b)(s-c)}$ where $s = \dfrac{a+b+c}{2}$ | $a + b + c$ |
| Equilateral triangle, side $a$ | $\dfrac{\sqrt{3}}{4}a^2$ | $3a$ |
| Right triangle | $\dfrac{1}{2} \times \text{base} \times \text{height}$ | Sum of the three sides |
| Parallelogram | $bh$ | $2(a + b)$ |
| Rhombus, diagonals $d_1, d_2$ | $\dfrac{1}{2}d_1 d_2$ | $4 \times \text{side}$ |
| Trapezium, parallel sides $a, b$ | $\dfrac{1}{2}(a + b)h$ | Sum of the four sides |
| Circle | $\pi r^2$ | $2\pi r$ |
| Sector, angle $\theta$ | $\dfrac{\theta}{360}\pi r^2$ | Arc $= \dfrac{\theta}{360} \times 2\pi r$ |
| Ring (annulus) | $\pi(R^2 - r^2)$ | $2\pi(R + r)$ |
Three-dimensional solids:
| Solid | Volume | Curved surface | Total surface |
|---|---|---|---|
| Cube, edge $a$ | $a^3$ | $4a^2$ | $6a^2$ |
| Cuboid $l, b, h$ | $lbh$ | $2h(l + b)$ | $2(lb + bh + hl)$ |
| Cylinder | $\pi r^2 h$ | $2\pi r h$ | $2\pi r(r + h)$ |
| Cone, slant $l$ | $\dfrac{1}{3}\pi r^2 h$ | $\pi r l$ | $\pi r(r + l)$ |
| Sphere | $\dfrac{4}{3}\pi r^3$ | not applicable | $4\pi r^2$ |
| Hemisphere | $\dfrac{2}{3}\pi r^3$ | $2\pi r^2$ | $3\pi r^2$ |
Key relations:
$$\text{Diagonal of a square} = a\sqrt{2} \qquad \text{Diagonal of a cuboid} = \sqrt{l^2 + b^2 + h^2} \qquad \text{Cube diagonal} = a\sqrt{3}$$
$$l = \sqrt{r^2 + h^2} \qquad \text{Sum of interior angles of an } n\text{-gon} = (n - 2) \times 180^\circ$$
| Fact | Value |
|---|---|
| Angles on a straight line | $180^\circ$ |
| Angles around a point | $360^\circ$ |
| Angles of a triangle | Sum is $180^\circ$ |
| Exterior angle of a triangle | Sum of the two opposite interior angles |
| Each interior angle of a regular $n$-gon | $\dfrac{(n - 2) \times 180}{n}$ |
| Pythagorean triples | 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41 and their multiples |
| Unit conversions | $1\ \text{m}^2 = 10000\ \text{cm}^2$, $1\ \text{m}^3 = 1000$ litres, $1\ \text{m}^3 = 10^6\ \text{cm}^3$ |
Part 3 — Shortcut methods and when they fail
| Shortcut | How it works | When it fails |
|---|---|---|
| π = 22/7 | Cancels cleanly when the radius is 7, 14, 21, 3.5 | Gives a slightly different value from 3.14. When options are close or the radius is not a multiple of 7, use the form the question gives |
| Pythagorean triples | Skip the square root | The sides are not always a triple. Check with squares if in doubt |
| Scaling rule | Lengths ×k, areas ×k², volumes ×k³ | Only for similar shapes. Stretching one dimension changes things differently |
| Percentage change in area | $a + b + ab/100$ for two sides | For a circle, radius up p% means area up $2p + p^2/100$ per cent, the same pattern |
| Recasting | Volume before = volume after | Surface area is not conserved. Only volume is |
| Half the diameter first | Removes the largest source of errors | None, but check whether the question gives the radius or the diameter |
| Path or border area | Outer area minus inner area | For a path inside a rectangle subtract from both length and width, and for a path outside add to both |
| Hemisphere formulas | Take half the sphere and add the flat circle | The curved surface is $2\pi r^2$, but the total surface includes the circular face: $3\pi r^2$ |
Unit rule. Convert all lengths to one unit before you start. Mixing metres and centimetres inside a single formula is the commonest source of an answer that is off by a factor of 100 or 10000.
Two-step recasting. When one solid is melted into another, equate the volumes, cancel π and common factors, and only then plug in numbers. The arithmetic collapses quickly.
Part 4 — Worked examples: flat figures
Worked example 1: rectangle. A rectangle has sides 12 and 5. Find its area, perimeter and diagonal.
Area $= 12 \times 5 = 60$. Perimeter $= 2(12 + 5) = 34$. Diagonal $= \sqrt{144 + 25} = \sqrt{169} = 13$, which is the 5-12-13 triple.
Worked example 2: Heron's formula. Find the area of a triangle with sides 13, 14 and 15.
| Step | What we do | Result |
|---|---|---|
| 1 | Semi-perimeter | $s = \dfrac{13 + 14 + 15}{2} = 21$ |
| 2 | Differences | $21 - 13 = 8$, $21 - 14 = 7$, $21 - 15 = 6$ |
| 3 | Area | $\sqrt{21 \times 8 \times 7 \times 6} = \sqrt{7056} = 84$ |
Check using the base 14: the height is $\dfrac{2 \times 84}{14} = 12$, which splits the triangle into a 5-12-13 and a 9-12-15 right triangle.
Worked example 3: circle and wheel. (a) A circle has radius 7. Find its area and circumference. (b) A wheel has diameter 70 cm. How many revolutions does it make in 1.1 km?
(a) Area $= \dfrac{22}{7} \times 49 = 154$ and circumference $= 2 \times \dfrac{22}{7} \times 7 = 44$. (b) Circumference $= \dfrac{22}{7} \times 70 = 220$ cm. The distance is $1.1\ \text{km} = 110000$ cm, so revolutions $= \dfrac{110000}{220} = 500$.
Common wrong step: putting 70 into $\pi r^2$ or $2\pi r$ without halving. The diameter is 70, so r = 35.
Worked example 4: equilateral triangle. Find the area and height of an equilateral triangle of side 6.
Area $= \dfrac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \approx 15.6$. Height $= \dfrac{\sqrt{3}}{2} \times 6 = 3\sqrt{3}$. Check: $\dfrac{1}{2} \times 6 \times 3\sqrt{3} = 9\sqrt{3}$.
Worked example 5: trapezium. The parallel sides of a trapezium are 10 and 14 and the distance between them is 8. Find the area.
$\dfrac{1}{2}(10 + 14) \times 8 = 12 \times 8 = 96$.
Worked example 6: path around a park. A rectangular park is 20 m by 15 m. A path 2 m wide runs around the outside. Find the area of the path.
| Step | What we do | Result |
|---|---|---|
| 1 | Outer rectangle adds 2 m on each side | $24 \times 19 = 456$ |
| 2 | Inner rectangle | $20 \times 15 = 300$ |
| 3 | Path | $456 - 300 = 156\ \text{m}^2$ |
Common wrong step: adding only 2 to each dimension to get 22 × 17. The path adds 2 m on both ends of each side, so each dimension grows by 4.
Worked example 7: ring. A circular track has an inner radius 14 m and a width of 7 m. Find its area.
Outer radius $= 21$ m. Area $= \dfrac{22}{7}(21^2 - 14^2) = \dfrac{22}{7}(441 - 196) = \dfrac{22}{7} \times 245 = 770\ \text{m}^2$.
Worked example 8: sector. A sector of a circle of radius 21 has an angle of $60^\circ$. Find its arc length and area.
Arc $= \dfrac{60}{360} \times 2 \times \dfrac{22}{7} \times 21 = \dfrac{1}{6} \times 132 = 22$. Area $= \dfrac{1}{6} \times \dfrac{22}{7} \times 441 = \dfrac{1}{6} \times 1386 = 231$.
Part 5 — Worked examples: solids and geometry
Worked example 9: cube and cuboid. (a) A cube has volume 343. Find its edge and total surface area. (b) A cuboid is 10 × 8 × 6. Find its volume, total surface area and longest diagonal.
(a) Edge $= \sqrt[3]{343} = 7$, TSA $= 6 \times 49 = 294$. (b) Volume $= 480$. TSA $= 2(80 + 48 + 60) = 376$. Diagonal $= \sqrt{100 + 64 + 36} = \sqrt{200} = 10\sqrt{2} \approx 14.1$.
Worked example 10: cylinder. A cylinder has r = 7 and h = 10. Find its volume, curved surface area and total surface area.
Volume $= \dfrac{22}{7} \times 49 \times 10 = 1540$. CSA $= 2 \times \dfrac{22}{7} \times 7 \times 10 = 440$. TSA $= 440 + 2 \times 154 = 748$, or directly $2\pi r(r + h) = 44 \times 17 = 748$.
Worked example 11: cone. A cone has r = 3 and h = 4. Find the slant height, volume and curved surface area.
| Step | What we do | Result |
|---|---|---|
| 1 | Slant height | $\sqrt{9 + 16} = 5$ |
| 2 | Volume | $\dfrac{1}{3} \times \dfrac{22}{7} \times 9 \times 4 = \dfrac{264}{7} \approx 37.7$ |
| 3 | CSA | $\dfrac{22}{7} \times 3 \times 5 = \dfrac{330}{7} \approx 47.1$ |
Common wrong step: using h = 4 in the CSA formula. The curved surface uses the slant height l = 5.
Worked example 12: sphere. Find the volume of a sphere of radius 21.
$\dfrac{4}{3} \times \dfrac{22}{7} \times 21^3 = \dfrac{4}{3} \times \dfrac{22}{7} \times 9261 = \dfrac{4}{3} \times 29106 = 38808$.
Worked example 13: recasting. A solid metal sphere of radius 6 cm is melted and made into small spheres of radius 2 cm. How many small spheres are formed?
Volumes are in the ratio of the cubes of the radii, so the number is $\left(\dfrac{6}{2}\right)^3 = 27$. Check: $\dfrac{4}{3}\pi \times 216$ divided by $\dfrac{4}{3}\pi \times 8$ is 27.
Worked example 14: scaling. The radius of a sphere is doubled. What happens to its surface area and volume?
Surface area scales as the square of the radius, so it becomes 4 times as large. Volume scales as the cube, so it becomes 8 times as large. A rise in the edge of a cube by 10% raises the volume by $1.1^3 - 1 = 0.331$, which is 33.1%.
Worked example 15: ladder and tank. (a) A 13 m ladder reaches a point 12 m up a wall. How far is its foot from the wall? (b) A tank is 5 m long, 4 m wide and 3 m deep. How many litres does it hold?
(a) Distance $= \sqrt{13^2 - 12^2} = \sqrt{25} = 5$ m. (b) Volume $= 5 \times 4 \times 3 = 60\ \text{m}^3$, and $1\ \text{m}^3 = 1000$ litres, so it holds 60,000 litres.
Part 6 — Common traps
| Trap | Why it is tempting | The truth |
|---|---|---|
| Using the diameter as the radius | The diameter is the number given | Halve the diameter before any circle formula |
| Square units for volume or cubic units for area | The formula letters look similar | Area is in square units, volume in cubic units |
| Using the vertical height in the cone CSA | The height is given | CSA uses the slant height $l = \sqrt{r^2 + h^2}$ |
| Hemisphere total surface as $2\pi r^2$ | That is the curved part | Total surface of a solid hemisphere is $3\pi r^2$, because it includes the flat circular face |
| Doubling a side doubles the area | Linear thinking | Area ×4, volume ×8 |
| Mixing cm and m | Both appear in the question | Convert to a single unit first |
| Path width added once | Each side seems to need only one width | A path outside adds the width twice to each dimension. A path inside subtracts it twice |
| Assuming 22/7 is exact | It is the usual value | Use 3.14 if the question says so, or if the radius is not a multiple of 7 |
Part 7 — Rapid-fire practice with answers
| No. | Question | Answer |
|---|---|---|
| 1 | Area of a square whose diagonal is $8\sqrt{2}$ | 64 |
| 2 | Perimeter of a 15 by 9 rectangle | 48 |
| 3 | Area of a triangle with base 12 and height 9 | 54 |
| 4 | A right triangle has legs 6 and 8. Hypotenuse and area | 10 and 24 |
| 5 | Area of a circle of diameter 14 | 154 |
| 6 | Circumference of a circle of radius 3.5 | 22 |
| 7 | Rhombus with diagonals 10 and 24. Area and side | 120 and 13 |
| 8 | Cube of edge 4. Volume and total surface area | 64 and 96 |
| 9 | Volume of a 6 × 5 × 4 cuboid | 120 |
| 10 | Volume of a cylinder with r = 3.5 and h = 10 | 385 |
| 11 | Surface area of a sphere of radius 7 | 616 |
| 12 | Total surface area of a solid hemisphere of radius 7 | 462 |
| 13 | Area of an equilateral triangle of side 4 | $4\sqrt{3}$ |
| 14 | Radius doubles. Change in the area of a circle | ×4 |
| 15 | Edge of a cube doubles. Change in volume | ×8 |
| 16 | Interior angle of a regular octagon | 135° |
| 17 | Diagonal of a square of side 5 | $5\sqrt{2}$ |
Part 8 — A timed-practice routine
| Day | Task | Time |
|---|---|---|
| 1 | Write both formula tables from memory, then compare with Part 2 | 15 min |
| 2 | Re-solve worked examples 1 to 8 closed book, writing the radius separately from the diameter | 25 min |
| 3 | Re-solve worked examples 9 to 15 closed book, and sketch each solid before you start | 25 min |
| 4 | Rapid-fire table against the clock: 17 questions in 12 minutes | 12 min |
| 5 | Mixed set of 10 questions at 60 seconds each, marked as solved, guessed or skipped | 10 min |
| 6 | Redo every miss and write which unit or which radius caused it | 20 min |
| 7 | Timed test of 15 questions in 15 minutes, with accuracy first | 15 min |
In the exam. Write the formula first, then substitute. Underline the unit the answer must be in. If two options differ by a factor of 2, 4 or 8, you have probably missed a diameter, a squared unit or a cubed unit.
Part 9 — Where this connects
The scaling rule is the percentage and successive-change idea from Percentage, Ratio and Alligation, and tank filling meets rates in Time, Speed and Work. Shapes unfold and fold in Non-Verbal Reasoning. The ideas of lines, circles and distances in a plane are developed fully in Straight Lines and Conic Sections, and finding the best dimensions of a solid is done with calculus in Application of Derivatives. Areas under curves are the subject of Application of Integrals.
One-line revision: area is in square units and volume in cubic units, halve the diameter before using a circle or sphere formula, the slant height of a cone is the square root of r squared plus h squared, π is 22/7, and doubling every length multiplies area by 4 and volume by 8.
Practice questions
Now test yourself. 8 free sample questions with explanations. 27 more in the timed practice test.
Q1. A solid metal cube of side $6$ cm is melted and recast into small cubes of side $2$ cm. How many small cubes are formed?
▸ Show answer & explanation
Answer: A
Big cube volume $=6^3=216 \text{ cm}^3$. Small cube volume $=2^3=8 \text{ cm}^3$. Number $=\frac{216}{8}=27$.
Q2. The length of a rectangle is twice its breadth and its area is $72 \text{ cm}^2$. What is its perimeter?
▸ Show answer & explanation
Answer: A
Let breadth $=b$, length $=2b$. Area $=2b^2=72$, so $b^2=36$, $b=6$, $l=12$. Perimeter $=2(12+6)=36 \text{ cm}$.
Q3. A wire is bent into a circle of radius $14$ cm. If the same wire is re-bent into a square, what is the side of the square? (Use $\pi=\frac{22}{7}$.)
▸ Show answer & explanation
Answer: A
Length of wire $=$ circumference $=2\pi r=2\times\frac{22}{7}\times14=88$ cm. Side of square $=\frac{88}{4}=22 \text{ cm}$.
Q4. In a triangle, two angles measure $55^\circ$ and $65^\circ$. What is the exterior angle at the third vertex?
▸ Show answer & explanation
Answer: A
An exterior angle equals the sum of the two remote interior angles: $55^\circ+65^\circ=120^\circ$.
Q5. A circle is inscribed in a square of side $14$ cm. What is the area of the circle? (Use $\pi=\frac{22}{7}$.)
▸ Show answer & explanation
Answer: A
An inscribed circle has diameter equal to the side, so radius $=7$ cm. Area $=\pi r^2=\frac{22}{7}\times49=154 \text{ cm}^2$.
Q6. The total surface area of a cube is $216 \text{ cm}^2$. What is its volume?
▸ Show answer & explanation
Answer: A
Total surface area $=6s^2=216$, so $s^2=36$, $s=6$ cm. Volume $=s^3=6^3=216 \text{ cm}^3$.
Q7. Two parallel lines are cut by a transversal. If one of the co-interior (allied) angles is $70^\circ$, what is the measure of the other co-interior angle on the same side?
▸ Show answer & explanation
Answer: A
Co-interior angles between parallel lines are supplementary, summing to $180^\circ$. So the other angle $=180^\circ-70^\circ=110^\circ$.
Q8. A rectangular sheet of paper $44$ cm by $20$ cm is rolled along its longer side to form a cylinder. What is the radius of the cylinder? (Use $\pi=\frac{22}{7}$.)
▸ Show answer & explanation
Answer: A
Rolling along the $44$ cm side makes the circumference $44$ cm: $2\pi r=44$, so $r=\frac{44\times7}{2\times22}=7 \text{ cm}$.
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