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 & Basic Geometry asks you to compute area, perimeter, surface area and volume of 2-D figures and 3-D solids, plus a few core geometry facts (angles, triangles, Pythagoras).
- Almost every mark here is a formula correctly recalled and applied with the right units — area is in square units, volume in cubic units, and the commonest mistake of all is mixing them or confusing radius with diameter.
- This guide gives you the full 2-D and 3-D formula set with labelled diagrams, the geometry essentials, and worked examples.
- Exam reality: +5 / −1. Keep π = 22/7 (or 3.14) handy and watch the units.
Part 1 — Two-dimensional figures
| Figure | Area | Perimeter |
|---|---|---|
| Square (side a) | a² | 4a |
| Rectangle (l, b) | l × b | 2(l + b) |
| Triangle (base b, height h) | ½ × b × h | a + b + c |
| Triangle (sides a,b,c) | √(s(s−a)(s−b)(s−c)), s = ½(a+b+c) | a + b + c |
| Equilateral (side a) | (√3 ÷ 4) a² | 3a |
| Parallelogram (b, h) | b × h | 2(a + b) |
| Trapezium (parallel a, b; height h) | ½ (a + b) × h | sum of sides |
| Rhombus (diagonals d₁, d₂) | ½ × d₁ × d₂ | 4 × side |
| Circle (radius r) | π r² | 2 π r (circumference) |
The diagonal of a square is a√2; of a rectangle, √(l² + b²).
Part 2 — Three-dimensional solids
| Solid | Volume | Surface area |
|---|---|---|
| Cube (side a) | a³ | 6a² |
| Cuboid (l, b, h) | l × b × h | 2(lb + bh + hl) |
| Cylinder (r, h) | π r² h | CSA 2πrh · TSA 2πr(r+h) |
| Cone (r, h, slant l) | ⅓ π r² h | CSA πrl · TSA πr(r+l) |
| Sphere (r) | (4/3) π r³ | 4 π r² |
| Hemisphere (r) | (2/3) π r³ | CSA 2πr² · TSA 3πr² |
For a cone, the slant height l = √(r² + h²). CSA = curved surface area; TSA = total (includes the flat faces).
Part 3 — Basic geometry essentials
- Angles on a straight line sum to 180°; around a point, 360°; the three angles of a triangle sum to 180°.
- Pythagoras: in a right triangle, hypotenuse² = base² + height². Memorise the triples 3-4-5, 5-12-13, 8-15-17, 7-24-25 — they appear constantly.
- An exterior angle of a triangle equals the sum of the two opposite interior angles.
- Area of a triangle can also be written ½ × base × height or via Heron's formula when you only know the three sides.
Part 4 — Key constants & speed techniques
- π = 22/7 is exact enough; it cancels cleanly when the radius is a multiple of 7.
- Spot the Pythagorean triple to skip the square-root step.
- Units: area is square, volume is cubic — a "double the side" change multiplies area by 4 and volume by 8.
- Half the diameter before using any circle/sphere formula — radius, not diameter.
- Match units first (convert cm to m, etc.) before computing.
Part 5 — Worked examples
1. Rectangle. l = 12, b = 5. Area = 60 sq units; diagonal = √(144+25) = √169 = 13.
2. Triangle (Heron). Sides 13, 14, 15. s = 21; area = √(21·8·7·6) = √7056 = 84.
3. Circle. r = 7. Area = 22/7 × 49 = 154; circumference = 2 × 22/7 × 7 = 44.
4. Equilateral. Side 6. Area = (√3/4) × 36 = 9√3.
5. Cube. Side 5. Volume = 125; surface area = 6 × 25 = 150.
6. Cylinder. r = 7, h = 10. Volume = 22/7 × 49 × 10 = 1540; CSA = 2 × 22/7 × 7 × 10 = 440.
7. Cone. r = 3, h = 4 → slant = 5. Volume = ⅓ × 22/7 × 9 × 4 = 37.7; CSA = 22/7 × 3 × 5 = 47.1.
8. Sphere. r = 21. Volume = 4/3 × 22/7 × 21³ = 38,808.
Part 6 — Common traps
- Square vs cubic units — area never carries cubic units; volume never square.
- Radius vs diameter — halve the diameter first.
- CSA vs TSA — "curved" excludes the flat top/bottom; "total" includes them.
- Slant height ≠ vertical height for a cone — use l = √(r² + h²).
- Doubling a dimension multiplies area ×4 and volume ×8, not ×2.
Part 7 — How to use this page
Memorise the two formula tables and the Pythagorean triples, re-solve the eight examples closed-book, then attempt the practice set and the timed test.
One-line revision: area is square units and volume cubic, use radius not diameter, π = 22/7, slant height of a cone is √(r²+h²), and doubling a side 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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