Circle Geometry: Key Properties and Formulas for Placements

Circle geometry appears in the quantitative section of every major placement aptitude test in India, and the same theorem families account for nearly all the questions asked. Whether you’re sitting a campus placement evaluation test, a TCS NQT, or an off-campus drive assessment, the geometry section draws from chord properties, tangent rules, inscribed angles, and cyclic quadrilaterals. This article covers the definitions, the theorems, and three worked examples built around the question formats that appear most in actual tests. If you want a structured resource list alongside this, see the guide to best books for placement preparation.

Core Definitions and Formulas

Before the theorems, a quick-reference table for the terms placement tests assume you know. These definitions are formalised in NCERT Class 10 Mathematics, Chapter 10, the standard Indian curriculum source for circle geometry.

Term	Definition
Radius (r)	Distance from the centre O to any point on the circle
Diameter (d)	Longest chord; passes through the centre; d = 2r
Chord	Line segment with both endpoints on the circle
Arc	Portion of the circle’s circumference between two points
Tangent	Line touching the circle at exactly one point
Secant	Line intersecting the circle at two distinct points

The two formulas you will use in nearly every circle problem:

• Area of a circle: A = π × r²
• Circumference: C = 2πr

Where r is the radius and π ≈ 3.14159. Diameter d = 2r, so circumference can also be written as C = πd.

Chord Properties

A chord is a line segment with both endpoints on the circle. Three chord properties appear repeatedly in aptitude tests.

Perpendicular from Centre Bisects the Chord

The perpendicular drawn from the centre to any chord bisects that chord into two equal halves. This generates most “find the chord length” or “find the distance from the centre” questions. Given radius r and perpendicular distance d from the centre to the chord, the half-chord length equals √(r² - d²) and the full chord length is 2√(r² - d²). This follows directly from the Pythagorean theorem applied to the right triangle formed by the radius, the perpendicular, and the half-chord.

Equal Chords are Equidistant from the Centre

If two chords in a circle are equal in length, they are equidistant from the centre. Conversely, chords at equal distances from the centre have equal lengths. Problems use this property in both directions: given equal lengths to conclude equal distances, or given equal distances to conclude equal lengths.

Equal Chords Subtend Equal Angles

Two equal chords in the same circle subtend equal angles at the centre. This property appears in problems where you’re asked to prove or calculate central angles without having all sides of the triangle.

Tangent Properties

A tangent is a line that touches the circle at exactly one point. It never crosses the circle. Four tangent properties appear in placement tests, ranging from foundational to higher-difficulty.

Tangent is Perpendicular to the Radius

At the point of contact, the tangent is always perpendicular to the radius drawn to that point. This 90-degree angle is the starting point for every tangent-length calculation. It creates a right-angled triangle whenever you connect the external point to the centre.

Two Tangents from an External Point are Equal

If PA and PB are two tangents drawn from an external point P to a circle with centre O and radius r, then PA = PB. The tangent length is derived from the right triangle OAP: since OA is perpendicular to PA, the Pythagorean theorem gives PA² = PO² - OA², which simplifies to PA² = PO² - r². Taking the square root gives the tangent length directly.

The Alternate Segment Theorem

The angle between a tangent and a chord at the point of contact equals the inscribed angle subtending the same chord from the opposite arc. In other words, if you draw a tangent at point T and a chord TC, the angle between the tangent and TC on one side equals the angle inscribed in the arc on the other side. This theorem appears in higher-difficulty questions and in engineering entrance test geometry sections.

Common Tangents Between Two Circles

The number of common tangents depends on the relative position of the two circles:

Configuration	Common Tangents
One circle inside the other, no contact	0
One circle inside the other, internally tangent	1
Circles overlap at two points	2
Circles touch externally at one point	3
Circles are completely separate	4

Angle Theorems

Three angle results account for most of the “find the angle” questions in placement circle geometry. Knowing all three lets you classify any question quickly before computing.

Inscribed Angle Theorem

An inscribed angle is formed when two chords meet at a point on the circle. The inscribed angle is always half the central angle that subtends the same arc. If a chord creates a central angle of 80°, any inscribed angle looking at the same arc from the major arc side equals 40°. All inscribed angles subtending the same arc are equal to each other, regardless of where on the arc the vertex sits.

Angle in a Semicircle

An angle inscribed in a semicircle is always 90°. This is a direct consequence of the inscribed angle theorem: the diameter subtends a central angle of 180°, so the inscribed angle equals 180° divided by 2, which is 90°. In practice: if one side of an inscribed angle is the diameter of the circle, the angle at the circumference is a right angle.

Cyclic Quadrilateral

A cyclic quadrilateral has all four vertices on the circle. In any cyclic quadrilateral, opposite angles are supplementary: they add to 180°. If angle A = 70°, then angle C = 110°. If angle B = 115°, then angle D = 65°. This property holds for every cyclic quadrilateral without exception, and placement tests use it both to find missing angles and to check whether a quadrilateral can be inscribed in a circle.

Worked Examples

The following examples cover three of the most common question types. Each is formatted as a step-by-step list to show the calculation chain. For additional practice organised by theorem type, see Khan Academy’s circles module. The step-identification approach used here transfers to other quantitative topics as well: time and work problems follow the same structure of identifying the right relationship before computing.

Example 1: Find the Chord Length Given Radius and Distance

• Given: A circle with radius 5 cm. A chord is at a perpendicular distance of 3 cm from the centre.
• Step 1: The perpendicular from the centre bisects the chord, so you have a right triangle with hypotenuse = radius = 5 cm and one leg = perpendicular distance = 3 cm.
• Step 2: Half-chord = √(5² - 3²) = √(25 - 9) = √16 = 4 cm.
• Step 3: Full chord length = 2 × 4 = 8 cm.
• Check: 3² + 4² = 9 + 16 = 25 = 5². Correct.

Example 2: Find the Tangent Length from an External Point

• Given: A circle with radius 6 cm. An external point P is 10 cm from the centre O.
• Step 1: The radius OA is perpendicular to the tangent PA at the contact point A.
• Step 2: Triangle OAP is right-angled at A, with OA = 6 cm and OP = 10 cm.
• Step 3: PA² = OP² - OA² = 100 - 36 = 64.
• Step 4: PA = √64 = 8 cm.
• Note: The second tangent PB from P also equals 8 cm, by the equal-tangents property.

Example 3: Find the Inscribed Angle Given the Central Angle

• Given: A chord subtends a central angle of 140° at the centre. A point Q lies on the major arc.
• Step 1: Identify which arc Q is on. Q is on the major arc (the larger arc), so it looks at the chord across the minor arc.
• Step 2: By the inscribed angle theorem, angle at Q = central angle / 2 = 140° / 2 = 70°.
• Variant: If the question asks for the angle from the minor arc instead, that inscribed angle = (360° - 140°) / 2 = 220° / 2 = 110°. The two inscribed angles from opposite arcs always sum to 180°.

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