chapter

Geometry and Mensuration Notes for GATE DA

Geometry and Mensuration notes for GATE DA: 9 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

geometry and mensuration notes

Chapter Roadmap: Geometry and Mensuration

Chapter Roadmap: Geometry and Mensuration Your journey through spatial reasoning and measurement. Master these to decode any geometric configuration efficiently. 1. Circle Geometry and Chord-Angle Relations Current Topic. Central angles, arcs, and segment properties. 2. Tangents, Secants, and Circle Theorems Upcoming. Tangent properties, alternate segment theorem, and intersecting chords. 3. Area and Perimeter of 2D Figures Upcoming. Triangles, quadrilaterals, and composite shapes. 4. Surface Area and Volume of 3D Solids Upcoming. Prisms, cylinders, cones, and spheres with unit conversions. What You Will Master Ability to quickly decode geometric configurations, apply chord and tangent theorems, and compute measurements without getting trapped by similar-figure ratios or hidden isosceles triangles.

Circle Geometry and Chord-Angle Relations

Circle Geometry and Chord-Angle Relations A circle is the set of all points in a plane that are at a fixed distance (the radius) from a central point (the center). When we connect points on the circumference, we create: Chord: A line segment connecting any two points on the circle. Arc: A continuous portion of the circumference. Segment: The region bounded by a chord and an arc. The core of circle geometry lies in understanding how the angles formed by these elements relate to one another, specifically the relationship between the angle at the center and the angle at the circumference.

The Central Angle Theorem

The Central Angle Theorem The angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circumference. ∠ A O B = 2 \times ∠ A P B O is the center of the circle. A and B are points on the circumference defining the arc. P is any other point on the remaining circumference. Intuition: The center "sees" the arc from a position of maximum leverage, resulting in an angle exactly twice as large as any point on the boundary.

6 more cards in this chapter

Try a question

Answer it here to see how it works. Nothing is recorded until you sign in.

Question 1

Level 1: Warm-up

If an arc subtends an angle of 3 5^{\circ} at a point on the remaining part of the circumference, what is the angle subtended by the same arc at the center of the circle?

Question 2

Level 1: Warm-up

An arc of a circle subtends an angle of 14 0^{\circ} at the center. What is the angle subtended by the same arc at any point on the remaining part of the circumference?

Question 3

Level 1: Warm-up

A triangle is inscribed in a circle such that one of its sides is the diameter of the circle. What is the measure of the angle opposite to the diameter?

Question 4

Level 1: Warm-up

If a triangle is inscribed in a circle and one of its interior angles is exactly 9 0^{\circ}, what must be true about the side opposite to this angle?

Question 5

Level 1: Warm-up

Points A, B, and C lie on a circle with center O . If the angle subtended by arc A C at point B on the circumference is 3 5^{\circ}, what is the measure of the central angle ∠ A O C in degrees?

Question 6

Level 1: Warm-up

Consider the following statements regarding circle geometry: Assertion (A): The angle subtended by a diameter at any point on the circumference is 9 0^{\circ} . Reason (R): The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circumference.

Question 7

Level 1: Warm-up

A continuous portion of the circumference of a circle is formally known as a:

Question 8

Level 1: Warm-up

Points A, B, C, and D lie on a circle. If ∠ A C B = 4 0^{\circ} and both C and D are on the same side of the chord A B, what is the measure of ∠ A D B ?

Question 9

Level 1: Warm-up

The region bounded by a chord and its corresponding arc in a circle is formally known as a:

Question 10

Level 1: Warm-up

For two angles subtended by the same chord to be exactly equal, what condition must be met regarding their positions?

Free preview ends here

Login to view the complete notes

Creating an account is free. You get the rest of this chapter, step-by-step solutions, and a study plan built around the topics you are actually weak at.

Why MastersUp

Personalised first. High quality throughout.

Most platforms hand everyone the same content. Here the content moves with your performance, topic by topic.

Built around you, not around a syllabus PDF

Every answer you give moves your topic-level intelligence rate. The next question, the next revision card and tomorrow's plan all change with it.

Revision that hits your weak spots

We only revise topics you have actually attempted and are still below the safe bar on — never the same chapter on repeat.

Questions calibrated to the real exam

Each question carries a measured toughness. You are served a rung above your current level, so practice keeps stretching you.

Notes written for recall, not for volume

Full lesson cards for first study, curated short-note cards for the last mile — with derivations, traps and exam patterns marked.

One place for everything

Notes, chapter practice, previous-year questions, test series and full-length papers — all feeding one picture of your preparation.

Honest progress

No vanity streaks. Progress here means chapters mastered and accuracy that held up on harder questions.

Unlock the whole course

Full notes and short notes, the complete question bank with worked solutions, mock tests, full-length papers, and an adaptive plan that rebuilds itself as you improve.

View all plans Start free

Geometry and Mensuration Notes for GATE DA

Geometry and Mensuration notes for GATE DA: 9 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Geometry and Mensuration

Your journey through spatial reasoning and measurement. Master these to decode any geometric configuration efficiently.

1. Circle Geometry and Chord-Angle Relations

Current Topic. Central angles, arcs, and segment properties.

2. Tangents, Secants, and Circle Theorems

Upcoming. Tangent properties, alternate segment theorem, and intersecting chords.

3. Area and Perimeter of 2D Figures

Upcoming. Triangles, quadrilaterals, and composite shapes.

4. Surface Area and Volume of 3D Solids

Upcoming. Prisms, cylinders, cones, and spheres with unit conversions.

What You Will Master

Ability to quickly decode geometric configurations, apply chord and tangent theorems, and compute measurements without getting trapped by similar-figure ratios or hidden isosceles triangles.

Circle Geometry and Chord-Angle Relations

A circle is the set of all points in a plane that are at a fixed distance (the radius) from a central point (the center).

When we connect points on the circumference, we create:

Chord: A line segment connecting any two points on the circle.
Arc: A continuous portion of the circumference.
Segment: The region bounded by a chord and an arc.

The core of circle geometry lies in understanding how the angles formed by these elements relate to one another, specifically the relationship between the angle at the center and the angle at the circumference.

The Central Angle Theorem

The angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circumference.

∠ A O B = 2 \times ∠ A P B

O is the center of the circle.

A and B are points on the circumference defining the arc.

P is any other point on the remaining circumference.

Intuition: The center "sees" the arc from a position of maximum leverage, resulting in an angle exactly twice as large as any point on the boundary.

Angles in the Same Segment

Angles subtended by the same arc (or chord) in the same segment of a circle are equal.

∠ A P B = ∠ A QB

Why this works: Both angles are subtended by the same arc

A B

. By the Central Angle Theorem, both are exactly half of the central angle

∠ A O B

. Therefore, they must be equal to each other.

Exam Application: This is a primary tool for "transferring" a known angle from one part of a complex diagram to another without any calculation.

Geometry and Mensuration: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Quantitative Aptitude MCQ

If an arc subtends an angle of $3 5^{\circ}$ at a point on the remaining part of the circumference, what is the angle subtended by the same arc at the center of the circle?

A.
$3 5^{\circ}$
B.
$5 5^{\circ}$
C.
$7 0^{\circ}$
D.
$10 5^{\circ}$

Correct Answer:

Step-by-Step Solution

Insight: The central angle is always exactly double the angle at the circumference for the same arc.

Exam route: Multiply the given circumference angle by 2: $3 5^{\circ} \times 2 = 7 0^{\circ}$ .

Learning route: By the Central Angle Theorem, the angle subtended by an arc at the center ( $∠ A O B$ ) is double the angle subtended by the same arc at any point on the remaining circumference ( $∠ A P B$ ). Therefore, $∠ A O B = 2 \times 3 5^{\circ} = 7 0^{\circ}$ .

Answer: C

Question 2 · Quantitative Aptitude MCQ

An arc of a circle subtends an angle of $14 0^{\circ}$ at the center. What is the angle subtended by the same arc at any point on the remaining part of the circumference?

A.
$7 0^{\circ}$
B.
$4 0^{\circ}$
C.
$14 0^{\circ}$
D.
$28 0^{\circ}$

Correct Answer:

Step-by-Step Solution

Insight: The angle at the circumference is exactly half the angle at the center for the same arc.

Exam route: Divide the central angle by 2: $14 0^{\circ} /2 = 7 0^{\circ}$ .

Learning route: According to the Central Angle Theorem, the angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining circumference. Therefore, the circumference angle is half of the central angle. Calculation: $14 0^{\circ} /2 = 7 0^{\circ}$ .

Answer: A

Question 3 · Quantitative Aptitude MCQ

A triangle is inscribed in a circle such that one of its sides is the diameter of the circle. What is the measure of the angle opposite to the diameter?

A.
$4 5^{\circ}$
B.
$6 0^{\circ}$
C.
$9 0^{\circ}$
D.
$18 0^{\circ}$

Correct Answer:

Step-by-Step Solution

Insight: The angle subtended by a diameter at the circumference is always a right angle.

Exam route: Recall the "angle in a semicircle" theorem: it is always $9 0^{\circ}$ .

Learning route: A diameter subtends a straight angle ( $18 0^{\circ}$ ) at the center of the circle. By the Central Angle Theorem, the angle at the circumference is exactly half of the central angle. Therefore, $18 0^{\circ} /2 = 9 0^{\circ}$ .

Answer: C

Question 4 · Quantitative Aptitude MCQ

If a triangle is inscribed in a circle and one of its interior angles is exactly $9 0^{\circ}$ , what must be true about the side opposite to this angle?

A.
It is a radius of the circle
B.
It is a diameter of the circle
C.
It is a tangent to the circle
D.
It is equal to the radius

Correct Answer:

Step-by-Step Solution

Insight: The converse of the angle in a semicircle theorem guarantees that a $9 0^{\circ}$ inscribed angle subtends a diameter.

Exam route: Recall that an angle of $9 0^{\circ}$ at the circumference means the opposite chord passes through the center, making it a diameter.

Learning route: The Angle in a Semicircle theorem states that an angle subtended by a diameter at the circumference is $9 0^{\circ}$ . The converse is also true: if an inscribed angle is $9 0^{\circ}$ , the chord opposite to it must be a diameter of the circle, as it subtends a $18 0^{\circ}$ angle at the center.

Answer: B

Question 5 · Quantitative Aptitude NAT

Points $A$ , $B$ , and $C$ lie on a circle with center $O$ . If the angle subtended by arc $A C$ at point $B$ on the circumference is $3 5^{\circ}$ , what is the measure of the central angle $∠ A O C$ in degrees?

Correct Answer:

70.00

Step-by-Step Solution

Key idea: This is a direct application of the Central Angle Theorem.

Step 1: Identify the given angle. The angle at the circumference is $∠ A B C = 3 5^{\circ}$ .

Step 2: Apply the theorem. The angle subtended by an arc at the center is double the angle subtended at the remaining part of the circumference.

Step 3: Calculate the central angle. $∠ A O C = 2 \times ∠ A B C = 2 \times 3 5^{\circ} = 7 0^{\circ}$ .

Answer: 70.00

Question 6 · Quantitative Aptitude MCQ

Consider the following statements regarding circle geometry:

Assertion (A): The angle subtended by a diameter at any point on the circumference is $9 0^{\circ}$ .

Reason (R): The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circumference.

A.
Both A and R are true and R is the correct explanation of A
B.
Both A and R are true but R is NOT the correct explanation of A
C.
A is true but R is false
D.
A is false but R is true

Correct Answer:

Step-by-Step Solution

Key idea: This is an assertion-reason question testing the foundational theorems of circle geometry.

Step 1: Evaluate Assertion (A). A diameter subtends a $18 0^{\circ}$ angle at the center (a straight line). By the Central Angle Theorem, the angle at the circumference is half of $18 0^{\circ}$ , which is $9 0^{\circ}$ . So, A is true.

Step 2: Evaluate Reason (R). This is the exact statement of the Central Angle Theorem. So, R is true.

Step 3: Check the relationship. The truth of A is directly derived from R. Therefore, R is the correct explanation of A.

Answer: A

Question 7 · Quantitative Aptitude MCQ

A continuous portion of the circumference of a circle is formally known as a:

A.
Chord
B.
Arc
C.
Segment
D.
Sector

Correct Answer:

Step-by-Step Solution

Insight: This is a direct terminology recall question about the parts of a circle.

Exam route: Match the definition "continuous portion of the circumference" to the term "Arc".

Learning route: A chord is a line segment connecting two points. An arc is the curved portion of the circumference between those two points. A segment is the region bounded by a chord and an arc. A sector is the region bounded by two radii and an arc.

Answer: B

Question 8 · Quantitative Aptitude MCQ

Points $A, B, C,$ and $D$ lie on a circle. If $∠ A C B = 4 0^{\circ}$ and both $C$ and $D$ are on the same side of the chord $A B$ , what is the measure of $∠ A D B$ ?

A.
$4 0^{\circ}$
B.
$5 0^{\circ}$
C.
$8 0^{\circ}$
D.
$14 0^{\circ}$

Correct Answer:

Step-by-Step Solution

Insight: Angles subtended by the same arc in the same segment are equal.

Exam route: Since $C$ and $D$ are on the same side of chord $A B$ , $∠ A D B = ∠ A C B = 4 0^{\circ}$ .

Learning route: Both $∠ A C B$ and $∠ A D B$ are subtended by the same arc $A B$ and lie in the same segment (same side of the chord). By the Angles in the Same Segment theorem, these angles must be exactly equal. Therefore, $∠ A D B = 4 0^{\circ}$ .

Answer: A

Question 9 · Quantitative Aptitude MCQ

The region bounded by a chord and its corresponding arc in a circle is formally known as a:

A.
Sector
B.
Tangent
C.
Segment
D.
Secant

Correct Answer:

Step-by-Step Solution

Insight: This is a direct definition recall of circle parts.

Exam route: Match "region bounded by a chord and an arc" to the term "Segment".

Learning route: A circle is divided into specific regions by its lines. A "sector" is the region bounded by two radii and an arc. A "segment" is the region bounded by a chord and an arc. A "tangent" is a line touching the circle at one point, and a "secant" is a line intersecting the circle at two points.

Answer: C

Question 10 · Quantitative Aptitude MCQ

For two angles subtended by the same chord to be exactly equal, what condition must be met regarding their positions?

A.
They must be on opposite sides of the chord
B.
One must be at the center of the circle
C.
They must be supplementary
D.
They must be on the same side of the chord

Correct Answer:

Step-by-Step Solution

Insight: Angles subtended by the same chord are equal only if they are in the same segment.

Exam route: Recall that "same segment" means the points must be on the same side of the chord.

Learning route: The Angles in the Same Segment theorem states that angles subtended by the same arc or chord in the same segment are equal. The "same segment" condition strictly requires the vertices of the angles to be on the same side of the chord. If they are on opposite sides, the angles are supplementary, not equal.

Answer: D

geometry and mensuration notes

Chapter Roadmap: Geometry and Mensuration

Chapter Roadmap: Geometry and Mensuration

Circle Geometry and Chord-Angle Relations

Circle Geometry and Chord-Angle Relations

The Central Angle Theorem

The Central Angle Theorem

Try a question

Login to view the complete notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Geometry and Mensuration Notes for GATE DA

Chapter Roadmap: Geometry and Mensuration

Chapter Roadmap: Geometry and Mensuration

Circle Geometry and Chord-Angle Relations

Circle Geometry and Chord-Angle Relations

The Central Angle Theorem

The Central Angle Theorem

Angles in the Same Segment

Angles in the Same Segment

Geometry and Mensuration: Solved Questions with Step-by-Step Explanations (10 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

More notes in this unit