Competitive Exams / CUET UG / Geometry & Mensuration

Geometry & Mensuration for CUET UG

Master lines, angles, triangles, circles, polygons, and 2D plus 3D mensuration with concept-first notes, solved examples, revision shortcuts, and a timed practice route built in the same Learn at My Place UI flow.

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Overview

Why Geometry and Mensuration Matter in CUET

Geometry rewards theorem recognition and clean diagram reading. Mensuration rewards formula recall, unit control, and fast substitution. Together they create a strong scoring chapter for CUET aspirants who revise patterns instead of memorizing disconnected facts.

The exam usually mixes quick angle problems, triangle ratios, circle properties, area formulas, and volume scaling. That is why this module moves from core geometry into 2D and 3D mensuration rather than treating them as unrelated blocks.

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Section A

Notes & Concept Builder

Identify the figure, then the theorem

1. Lines, Angles, and Parallel Line Rules 2. Triangle Angle Facts and Exterior Angle 3. Congruency, Similarity, and BPT 4. Centroid, Incenter, Circumcenter, and Orthocenter 5. Pythagoras and Special Triangles 6. Circle Theorems and Cyclic Quadrilaterals 7. Quadrilaterals and Polygons 8. 2D Mensuration Formulas 9. 3D Mensuration and Scaling 10. Exam Strategy for Geometry and Mensuration

1. Lines, Angles, and Parallel Line Rules

Geometry begins with lines and the angle relations formed when lines intersect or stay parallel. In CUET, this section is less about drawing perfect figures and more about spotting standard angle patterns immediately.

\text{Vertically opposite angles are equal} \qquad \text{and} \qquad \text{a linear pair sums to } 180^\circ

When a transversal cuts parallel lines, corresponding angles are equal, alternate interior angles are equal, and co-interior angles are supplementary.

10-second shortcut: Think of the F, Z, and C patterns. F means corresponding equals, Z means alternate equals, and C means same-side interior adds to

180^\circ

2. Triangle Angle Facts and Exterior Angle

The three interior angles of a triangle always add to $180^\circ$ . The exterior angle of a triangle equals the sum of the two remote interior angles.

A + B + C = 180^\circ \qquad \text{and} \qquad \text{Exterior angle} = \text{sum of opposite interior angles}

This is one of the fastest scoring ideas in CUET because angle ratio questions reduce to a single linear equation.

3. Congruency, Similarity, and BPT

Congruent triangles have the same size and shape. Similar triangles have the same shape but may differ in size. CUET questions usually test whether corresponding angles match and whether side ratios stay equal.

\triangle ABC \sim \triangle DEF \Rightarrow \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}

The Basic Proportionality Theorem links parallel lines and similarity. If a line is parallel to one side of a triangle, it divides the other two sides proportionally.

Exam trap: SSA is not a standard congruency rule. Do not treat it as valid unless extra conditions are given.

4. Centroid, Incenter, Circumcenter, and Orthocenter

Each triangle has four famous centers. The centroid is the balance point, the incenter is equidistant from the sides, the circumcenter is equidistant from the vertices, and the orthocenter is the meeting point of the altitudes.

\text{If } G \text{ is centroid and } AD \text{ is a median, then } AG:GD = 2:1

For equilateral triangles, all four centers coincide. For obtuse triangles, the circumcenter and orthocenter lie outside the triangle.

5. Pythagoras and Special Triangles

Right triangles drive a huge portion of geometry and mensuration. Pythagoras, standard triplets, and special-angle triangles make calculation much faster.

a^2 + b^2 = c^2

Useful triplets include $3,4,5$ , $5,12,13$ , and $8,15,17$ . Also memorize the ratios for a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle and a $45^\circ$ - $45^\circ$ - $90^\circ$ triangle.

10-second shortcut: If the numbers look close to a known triplet, test it before expanding formulas. It often saves half the time.

6. Circle Theorems and Cyclic Quadrilaterals

Circle questions reward theorem recognition. The angle at the center is twice the angle at the circumference on the same arc. A tangent is perpendicular to the radius at the point of contact.

\angle AOB = 2\angle ACB \qquad \text{and} \qquad OT \perp PT

In cyclic quadrilaterals, opposite angles add to $180^\circ$ . Tangents drawn from the same external point are equal.

7. Quadrilaterals and Polygons

Parallelograms, rectangles, rhombi, and squares are best handled through diagonal properties and angle behavior. Regular polygons are easiest through their exterior-angle formula.

\text{Sum of interior angles of an } n\text{-gon} = (n-2) \times 180^\circ

\text{Each exterior angle of a regular } n\text{-gon} = \frac{360^\circ}{n}

8. 2D Mensuration Formulas

2D mensuration questions are mostly about formula recall plus one clean substitution. Square, rectangle, triangle, circle, sector, rhombus, trapezium, and annulus are the usual CUET shapes.

\text{Area of circle} = \pi r^2 \qquad \text{Circumference} = 2\pi r

\text{Area of trapezium} = \frac12(a+b)h \qquad \text{Area of rhombus} = \frac12 d_1 d_2

Heron's formula becomes useful when all three sides of a triangle are given but no height is provided.

9. 3D Mensuration and Scaling

For cubes, cuboids, cylinders, cones, spheres, hemispheres, and frustums, questions often compare total surface area, curved surface area, and volume. The most important big-picture idea is scaling.

\text{Volume scales as the cube of linear size} \qquad \text{and surface area scales as the square}

That means doubling the radius multiplies sphere volume by $8$ , not $2$ . Melting and recasting questions simply conserve volume.

10. Exam Strategy for Geometry and Mensuration

Do not treat this chapter as disconnected formulas. First identify the figure, then identify the theorem or formula, then substitute carefully with units. This order prevents most mistakes.

10-second shortcut: Angles and theorem questions should be solved before long mensuration calculations during the exam because they are usually faster and safer.

Always check whether the answer should be in degrees, square units, cubic units, or plain length. CUET often tests this basic discipline.

Solved Practice

Solved Examples

Try first, then open the reasoning

Exam Trap: Geometry mistakes usually come from applying the wrong theorem, while mensuration mistakes usually come from wrong units or forgetting to square or cube dimensions.

Example 1: Two co-interior angles are in the ratio $2:3$. Find the larger angle.

Co-interior angles sum to $180^\circ$ .

Let them be $2x$ and $3x$ . Then $5x = 180$ , so $x = 36$ and the larger angle is $108^\circ$ .

Example 2: An exterior angle of a triangle is $140^\circ$. If one remote interior angle is $60^\circ$, find the other.

Exterior angle equals the sum of the two remote interior angles.

So the required angle is $140^\circ - 60^\circ = 80^\circ$ .

Example 3: The angles of a triangle are in the ratio $2:3:5$. Find the angles.

Let the angles be $2x$ , $3x$ , and $5x$ .

Then $10x = 180$ , so $x = 18$ . The angles are $36^\circ$ , $54^\circ$ , and $90^\circ$ .

Example 4: Three parallel lines create intercepts $6$ and $9$ on one transversal. If the first intercept on another transversal is $8$, find the second.

Use proportional intercepts: $\frac{6}{9} = \frac{8}{x}$ .

So $6x = 72$ and $x = 12$ .

Example 5: Find the sum of interior angles of a decagon.

Use $(n-2) \times 180^\circ$ .

For $n=10$ , the sum is $8 \times 180^\circ = 1440^\circ$ .

Example 6: Each interior angle of a regular polygon is $150^\circ$. Find the number of sides.

Exterior angle $= 180^\circ - 150^\circ = 30^\circ$ .

Number of sides $= \frac{360^\circ}{30^\circ} = 12$ .

Example 7: Two similar triangles have perimeters in ratio $2:3$. If one side of the smaller is $10$ cm, find the corresponding side of the larger.

Corresponding side ratio equals perimeter ratio.

So the larger side is $10 \times \frac32 = 15$ cm.

Example 8: Areas of two similar triangles are in the ratio $49:81$. Find the ratio of corresponding sides.

Side ratio is the square root of the area ratio.

So the ratio is $7:9$ .

Example 9: In triangle $ABC$, $DE \parallel BC$, $AD = 4$, $DB = 6$, and $AE = 5$. Find $EC$.

By BPT, $\frac{AD}{DB} = \frac{AE}{EC}$ .

So $\frac46 = \frac5{EC}$ , giving $EC = 7.5$ .

Example 10: The centroid divides a median of length $18$ cm. Find the distance from the vertex to the centroid.

The centroid divides the median in the ratio $2:1$ from the vertex.

So the required distance is $\frac23 \times 18 = 12$ cm.

Example 11: In a right triangle with legs $8$ and $15$, find the hypotenuse.

Recognize the $8$ - $15$ - $17$ triplet, or apply Pythagoras.

The hypotenuse is $17$ .

Example 12: The midpoint segment in a triangle measures $9$ cm. Find the parallel third side.

Midpoint segment equals half the third side.

So the third side is $18$ cm.

Example 13: If $\angle ACB = 40^\circ$ is an inscribed angle, find the central angle subtending the same arc.

The central angle is twice the inscribed angle.

So the central angle is $80^\circ$ .

Example 14: A tangent and radius meet at a point on a circle. Find the angle between them.

A tangent is perpendicular to the radius at the point of contact.

So the angle is $90^\circ$ .

Example 15: In a cyclic quadrilateral, one angle is $75^\circ$. Find the opposite angle.

Opposite angles of a cyclic quadrilateral are supplementary.

So the opposite angle is $105^\circ$ .

Example 16: A rectangle has diagonal $13$ cm and one side $5$ cm. Find its area.

Other side $= \sqrt{13^2 - 5^2} = 12$ cm.

Area $= 5 \times 12 = 60$ cm $^2$ .

Example 17: A circle is inscribed in a square of side $14$ cm. Find the area of the circle using $\pi = \frac{22}{7}$.

The radius is half the side, so $r = 7$ cm.

Area $= \pi r^2 = \frac{22}{7} \times 49 = 154$ cm $^2$ .

Example 18: Use Heron's formula to find the area of a triangle with sides $13$, $14$, and $15$.

Semi-perimeter $s = 21$ .

Area $= \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84$ cm $^2$ .

Example 19: A sphere has radius doubled. By what factor does its volume change?

Volume scales as the cube of the radius.

So the factor is $2^3 = 8$ .

Example 20: A cone and cylinder have the same base and height. Compare their volumes.

Volume of cone $= \frac13 \pi r^2 h$ and volume of cylinder $= \pi r^2 h$ .

So cone : cylinder $= 1:3$ .

Example 21: A cube has total surface area $384$ cm$^2$. Find its volume.

$6a^2 = 384$ gives $a^2 = 64$ , so $a = 8$ .

Volume $= a^3 = 512$ cm $^3$ .

Example 22: A sphere of radius $3$ cm is dropped into a cylindrical vessel of radius $6$ cm. Find the rise in water level.

Rise $= \frac{\text{volume of sphere}}{\text{area of cylinder base}}$ .

So rise $= \frac{\frac43\pi 27}{\pi 36} = 1$ cm.

Example 23: A cuboid has dimensions $3$, $4$, and $12$ cm. Find its space diagonal.

Use 3D Pythagoras: diagonal $= \sqrt{3^2 + 4^2 + 12^2}$ .

That gives $\sqrt{169} = 13$ cm.

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Section B

The Test Zone

This chapter has a separate practice route with 4 sectional sessions of 10 questions each and a mixed 40-question mock. Every question runs on a 60-second timer so we train theorem recall, formula accuracy, and exam speed together.

Sectional Tests

4 focused sessions on lines and angles, triangle geometry, 2D mensuration, and 3D mensuration.

Open Sectional Tests

Full Mixed Mock

40 questions blending geometry and mensuration so we practice the chapter the way CUET typically mixes it.

Start Mixed Mock

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