Last Updated: April 2026
Geometry and Mensuration is one of the most important topics in the IPMAT Quantitative Aptitude section, consistently contributing 4-8 questions per paper in both IPMAT Indore and JIPMAT. With the right formula mastery and conceptual clarity, geometry can be a high-scoring area in IPMAT 2027. This guide covers all key theorems, formulas, and practice questions.
IPMAT Geometry — Exam Coverage
| Topic | IPMAT Questions (Approx) | Difficulty | Sub-topics |
|---|---|---|---|
| Triangles | 2-3 | Medium | Properties, congruence, similarity, area, special triangles |
| Circles | 2-3 | Medium-High | Tangents, chords, arc, sector, inscribed angles |
| Quadrilaterals | 1-2 | Medium | Parallelogram, rectangle, rhombus, trapezium |
| Mensuration (2D) | 1-2 | Easy-Medium | Area, perimeter of standard figures |
| Solid Geometry (3D) | 1-2 | Medium | Cube, cuboid, cylinder, cone, sphere, pyramid |
| Coordinate Geometry | 1-2 | Medium | Distance, midpoint, slope, equation of line |
Triangles — Complete Property Set
Basic Properties
- Sum of angles = 180°
- Exterior angle = sum of two non-adjacent interior angles
- Triangle inequality: Sum of any two sides > third side
- The largest side is opposite the largest angle
Area Formulas
| Method | Formula |
|---|---|
| Base-height | Area = (1/2) × base × height |
| Heron’s formula | Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2 |
| Two sides + included angle | Area = (1/2)ab sin C |
| Equilateral triangle | Area = (√3/4)a² |
Special Triangles — IPMAT Favourites
| Type | Sides Ratio | Angles | Key Property |
|---|---|---|---|
| 30-60-90 | 1 : √3 : 2 | 30°, 60°, 90° | Hypotenuse = 2 × shortest side |
| 45-45-90 | 1 : 1 : √2 | 45°, 45°, 90° | Isosceles right triangle |
| 3-4-5 (Pythagorean triple) | 3 : 4 : 5 | 37°, 53°, 90° | Classic right triangle |
| 5-12-13 | 5 : 12 : 13 | — | Another Pythagorean triple |
| 8-15-17 | 8 : 15 : 17 | — | Pythagorean triple |
Centers of Triangle
| Center | Definition | Property |
|---|---|---|
| Centroid (G) | Intersection of medians | Divides each median in 2:1 from vertex; always inside |
| Circumcenter (O) | Intersection of perpendicular bisectors | Equidistant from all vertices; can be outside (obtuse triangle) |
| Incenter (I) | Intersection of angle bisectors | Equidistant from all sides; always inside |
| Orthocenter (H) | Intersection of altitudes | Can be inside/outside/on the triangle |
Circles — Complete Formula Sheet
| Concept | Formula |
|---|---|
| Area | πr² |
| Circumference | 2πr |
| Length of arc | (θ/360°) × 2πr |
| Area of sector | (θ/360°) × πr² |
| Area of segment | Area of sector – Area of triangle |
| Tangent-radius angle | 90° always |
| Angle subtended at center = 2 × angle at circumference (same arc) | Central angle = 2 × inscribed angle |
| Angles in same segment | Equal |
| Angle in semicircle | 90° (Thales’ theorem) |
Solid Geometry — 3D Formulas
| Shape | Volume | Total Surface Area | Lateral SA |
|---|---|---|---|
| Cube (side a) | a³ | 6a² | 4a² |
| Cuboid (l×b×h) | lbh | 2(lb+bh+hl) | 2h(l+b) |
| Cylinder (r,h) | πr²h | 2πr(r+h) | 2πrh |
| Cone (r,h,l) | (1/3)πr²h | πr(r+l) | πrl |
| Sphere (r) | (4/3)πr³ | 4πr² | — |
| Hemisphere (r) | (2/3)πr³ | 3πr² | 2πr² |
Practice MCQs — IPMAT Geometry 2027
Q1. In triangle ABC, angle B = 90°, AB = 5 cm, BC = 12 cm. Find AC.
(A) 13 cm (B) 17 cm (C) 7 cm (D) 15 cm
Answer: (A) AC = √(5²+12²) = √(25+144) = √169 = 13 cm
Q2. The area of a circle inscribed in an equilateral triangle of side 6 cm is:
(A) π cm² (B) 3π cm² (C) 9π cm² (D) 2π cm²
Answer: (B) Inradius of equilateral triangle = a/(2√3) = 6/(2√3) = √3 cm; Area = π(√3)² = 3π cm²
Q3. A cylinder has radius 7 cm and height 10 cm. Its volume is: (use π = 22/7)
(A) 1540 cm³ (B) 1450 cm³ (C) 1640 cm³ (D) 1740 cm³
Answer: (A) V = πr²h = (22/7) × 49 × 10 = 22 × 7 × 10 = 1540 cm³
Q4. In a circle, a chord of length 16 cm is at a distance of 6 cm from the centre. The radius of the circle is:
(A) 8 cm (B) 10 cm (C) 12 cm (D) 14 cm
Answer: (B) Half-chord = 8 cm; radius² = 8² + 6² = 64+36 = 100; radius = 10 cm
Q5. The diagonal of a square is 10√2 cm. Its area is:
(A) 50 cm² (B) 100 cm² (C) 200 cm² (D) 150 cm²
Answer: (B) Diagonal = a√2; a = 10; Area = a² = 100 cm²
Frequently Asked Questions
How many geometry questions appear in IPMAT Indore?
IPMAT Indore (Quantitative Ability section) typically has 4-8 questions from Geometry and Mensuration combined. The exact count varies by year. In IPMAT 2024 and 2025, geometry questions included problems on similar triangles, circle theorems, and 3D mensuration. Prepare all standard formulas and practice with timed conditions since 40 QA questions in 40 minutes leaves little margin.
Is Coordinate Geometry part of IPMAT syllabus?
Yes, Coordinate Geometry (straight lines, distance formula, section formula) is part of IPMAT QA syllabus. However, it is less emphasized than classical geometry and mensuration. For IPMAT 2027, ensure you know the basics: distance between points, midpoint formula, slope of a line, equation of line in different forms, and collinearity conditions. JIPMAT also tests basic coordinate geometry.
Also study on IPM Gurukul: IPMAT Number System | IPMAT Data Interpretation | Free IPMAT Mock Test