Geometry and Mensuration
CSAT · Quantitative Aptitude
Introduction
CSAT geometry stays at school level: angles, triangles, circles, areas and volumes. The marks come from remembering formulas exactly and spotting Pythagorean triples.
Key concepts
Lines, angles and polygons
- Angles on a straight line add to 180°; around a point to 360°.
- Triangle angles add to 180°; an exterior angle equals the sum of the two opposite interior angles.
- Polygon with n sides: interior angles sum to (n − 2) × 180°; exterior angles always sum to 360°; each exterior angle of a regular polygon = 360°/n.
Triangles
Pythagoras: a² + b² = c² in a right triangle. Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29 and their multiples. Area = ½ × base × height; equilateral = (√3/4)a².
Areas and perimeters
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | l × b | 2(l + b) |
| Square | a² | 4a |
| Circle | πr² | 2πr |
| Parallelogram | base × height | 2(a + b) |
| Trapezium | ½(a + b)h | sum of sides |
Volumes and surfaces
| Solid | Volume | Surface |
|---|---|---|
| Cuboid | lbh | 2(lb + bh + hl); diagonal √(l² + b² + h²) |
| Cube | a³ | 6a²; diagonal a√3 |
| Cylinder | πr²h | curved 2πrh; total 2πr(r + h) |
| Cone | ⅓πr²h | curved πrl (l = slant height) |
| Sphere | (4/3)πr³ | 4πr² |
Scaling
If lengths change by x% and y%, area changes by x + y + xy/100 %. Scaling every length by k multiplies area by k² and volume by k³.
- Polygon interior sum = (n − 2) × 180°
- a² + b² = c²
- Circle: πr², 2πr
- Cylinder: πr²h, 2πrh
- Cuboid diagonal = √(l² + b² + h²)
Solved examples
Example 1
What is the circumference of a circle of radius 28 cm? (Take π = 22/7)
(a) 176 cm (b) 198 cm (c) 2464 cm (d) 352 cm
- Circumference = 2πr = 2 × 22/7 × 28 = 176.
Answer: (a) 176 cm
Example 2
What is the sum of the interior angles of a polygon with 10 sides?
(a) 360° (b) 1440° (c) 1800° (d) 1620°
- Sum = (n − 2) × 180° = (10 − 2) × 180° = 1440°.
Answer: (b) 1440°
Example 3
What is each interior angle of a regular polygon with 12 sides?
(a) 30° (b) 160° (c) 165° (d) 150°
- Each exterior angle = 360°/12 = 30°.
- Interior = 180° − 30° = 150°.
Answer: (d) 150°
Example 4
What is the curved surface area of a cylinder with radius 7 cm and height 20 cm? (π = 22/7)
(a) 1188 sq cm (b) 3080 sq cm (c) 880 sq cm (d) 440 sq cm
- CSA = 2πrh = 2 × 22/7 × 7 × 20 = 880 sq cm.
Answer: (c) 880 sq cm
Example 5
What is the volume of a cylinder with radius 3 1/2 cm and height 12 cm? (π = 22/7)
(a) 462 cu cm (b) 264 cu cm (c) 924 cu cm (d) 154 cu cm
- V = πr²h = 22/7 × 3 1/2² × 12 = 462 cu cm.
Answer: (a) 462 cu cm
Example 6
What is the volume of a cylinder with radius 14 cm and height 10 cm? (π = 22/7)
(a) 6160 cu cm (b) 880 cu cm (c) 12320 cu cm (d) 2053 1/3 cu cm
- V = πr²h = 22/7 × 14² × 10 = 6160 cu cm.
Answer: (a) 6160 cu cm
Example 7
What is the volume of a cylinder with radius 7 cm and height 10 cm? (π = 22/7)
(a) 440 cu cm (b) 1540 cu cm (c) 3080 cu cm (d) 513 1/3 cu cm
- V = πr²h = 22/7 × 7² × 10 = 1540 cu cm.
Answer: (b) 1540 cu cm
Example 8
The length of a rectangle is increased by 10% and its breadth is increased by 20%. What is the percentage change in its area?
(a) increase of 34% (b) decrease of 32% (c) increase of 32% (d) increase of 30%
- Area = l × b, so use the successive-change formula x + y + xy/100.
- = 10 + (20) + (10)(20)/100 = 32%.
Answer: (c) increase of 32%
Example 9
What is the circumference of a circle of radius 14 cm? (Take π = 22/7)
(a) 110 cm (b) 176 cm (c) 616 cm (d) 88 cm
- Circumference = 2πr = 2 × 22/7 × 14 = 88.
Answer: (d) 88 cm
Example 10
What is the length of the longest rod that fits inside a box 1 m × 4 m × 8 m?
(a) 8 m (b) 9 m (c) 5 m (d) 10 m
- Longest rod = body diagonal = √(1² + 4² + 8²) = √81 = 9 m.
Answer: (b) 9 m
- With π = 22/7, pick radii in multiples of 7 for clean answers.
- Recognise triples instantly – no square roots needed.
- Area/volume scale with k² and k³.
- Mixing radius and diameter.
- Using total surface area when the question asks for curved surface area.
- Forgetting unit conversions (1 m² = 10,000 cm²).
Practice set
Attempt all 12 questions (about 90 seconds each), then check the worked solutions below.
Q1. What is the length of the longest rod that fits inside a box 3 m × 4 m × 12 m?
(a) 7 m (b) 12 m (c) 13 m (d) 14 m
Q2. What is the circumference of a circle of radius 21 cm? (Take π = 22/7)
(a) 264 cm (b) 1386 cm (c) 154 cm (d) 132 cm
Q3. The length of a rectangle is increased by 30% and its breadth is increased by 25%. What is the percentage change in its area?
(a) increase of 55% (b) decrease of 62 1/2% (c) increase of 62 1/2% (d) increase of 64 1/2%
Q4. The length of a rectangle is increased by 10% and its breadth is increased by 10%. What is the percentage change in its area?
(a) increase of 21% (b) increase of 23% (c) decrease of 21% (d) increase of 20%
Q5. What is each interior angle of a regular polygon with 5 sides?
(a) 72° (b) 108° (c) 144° (d) 118°
Q6. What is the volume of a cylinder with radius 7 cm and height 15 cm? (π = 22/7)
(a) 770 cu cm (b) 2310 cu cm (c) 660 cu cm (d) 4620 cu cm
Q7. What is the sum of the interior angles of a polygon with 9 sides?
(a) 1440° (b) 1260° (c) 360° (d) 1620°
Q8. What is the total surface area of a cube of edge 10 cm?
(a) 400 sq cm (b) 1000 sq cm (c) 600 sq cm (d) 610 sq cm
Q9. What is the area of a circle of radius 35 cm? (Take π = 22/7)
(a) 7700 sq cm (b) 220 sq cm (c) 3872 sq cm (d) 3850 sq cm
Q10. The length of a rectangle is increased by 25% and its breadth is increased by 20%. What is the percentage change in its area?
(a) decrease of 50% (b) increase of 52% (c) increase of 45% (d) increase of 50%
Q11. The length of a rectangle is increased by 50% and its breadth is increased by 25%. What is the percentage change in its area?
(a) increase of 87 1/2% (b) decrease of 87 1/2% (c) increase of 75% (d) increase of 89 1/2%
Q12. How many cubes of edge 1 cm can be cut from a cuboid 14 cm × 6 cm × 2 cm?
(a) 168 (b) 248 (c) 182 (d) 22
Q1 – (c) 13 m. Longest rod = body diagonal = √(3² + 4² + 12²) = √169 = 13 m.
Q2 – (d) 132 cm. Circumference = 2πr = 2 × 22/7 × 21 = 132.
Q3 – (c) increase of 62 1/2%. Area = l × b, so use the successive-change formula x + y + xy/100. = 30 + (25) + (30)(25)/100 = 62 1/2%.
Q4 – (a) increase of 21%. Area = l × b, so use the successive-change formula x + y + xy/100. = 10 + (10) + (10)(10)/100 = 21%.
Q5 – (b) 108°. Each exterior angle = 360°/5 = 72°. Interior = 180° − 72° = 108°.
Q6 – (b) 2310 cu cm. V = πr²h = 22/7 × 7² × 15 = 2310 cu cm.
Q7 – (b) 1260°. Sum = (n − 2) × 180° = (9 − 2) × 180° = 1260°.
Q8 – (c) 600 sq cm. TSA = 6a² = 6 × 10² = 600 sq cm.
Q9 – (d) 3850 sq cm. Area = πr² = 22/7 × 35 × 35 = 3850.
Q10 – (d) increase of 50%. Area = l × b, so use the successive-change formula x + y + xy/100. = 25 + (20) + (25)(20)/100 = 50%.
Q11 – (a) increase of 87 1/2%. Area = l × b, so use the successive-change formula x + y + xy/100. = 50 + (25) + (50)(25)/100 = 87 1/2%.
Q12 – (a) 168. Volume = 14 × 6 × 2 = 168 cu cm; each cube is 1 cu cm.
- (n − 2)×180°; exterior 360°/n
- Triples 3-4-5, 5-12-13, 8-15-17
- Cylinder V = πr²h
- Scale: area k², volume k³
