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NDA Maths · Properties of Triangle

In-circle, Circumcircle & Regular Polygons

The incircle (radius r = Δ/s) sits inside touching all three sides; the circumcircle (radius R = abc/4Δ) passes through all three vertices — and a regular polygon's inscribed circle follows the same idea with a cotangent.

All Properties of Triangle notesNDA Maths strategy

Why this matters

Only 6 PYQs, but they pull in the circle formulas: inradius, circumradius, the central-angle relation, and the regular-polygon inradius. A couple are really cosine-rule problems on a labelled triangle, so the sine/cosine tools carry over.

Concept 1 of 2

Incircle, Circumcircle & the Central Angle

Intuition

Both special circles are tied to the area: the inradius is area over semi-perimeter, the circumradius is the product of sides over four times the area. And the angle a chord subtends at the centre is twice the angle it subtends at the circumference.

Definition

  • Inradius: r=Δsr = \dfrac{\Delta}{s} (area over semi-perimeter).
  • Circumradius: R=abc4Δ=a2sin⁡AR = \dfrac{abc}{4\Delta} = \dfrac{a}{2\sin A}.
  • Central vs inscribed angle: an arc subtends an angle at the centre that is twice the angle it subtends at any point on the circle: ∠BOC=2∠BAC\angle BOC = 2\angle BAC.
  • Chord length: a chord subtending angle θ\theta at the centre of a circle of radius RR has length 2Rsin⁡θ22R\sin\dfrac{\theta}{2}.

Inradius, circumradius, central angle

r=Δs,R=abc4Δ,∠BOC=2 ∠BACr = \dfrac{\Delta}{s}, \quad R = \dfrac{abc}{4\Delta}, \quad \angle BOC = 2\,\angle BAC
O, RI, rABCcircumradius R = abc/4Δ · inradius r = Δ/s

Worked example

A chord subtends an angle of 90∘90^\circ at the centre of a circle of radius RR. Find its length.
  1. Chord length =2Rsin⁡θ2= 2R\sin\dfrac{\theta}{2} with θ=90∘\theta = 90^\circ.
  2. =2Rsin⁡45∘=2R⋅12= 2R\sin 45^\circ = 2R\cdot\dfrac{1}{\sqrt2}.
Answer:R2R\sqrt{2}.

From the bank · past-year question

Example 1Properties of TriangleMODERATE
ABC is inscribed in a circle with centre O. Let α=∠BAC\alpha = \angle BAC, β=∠BOC\beta = \angle BOC. Which is correct?

[Q42 · Apr · 2018]

Drill 3 more on incircle, circumcircle & the central angle

Concept 2 of 2

Regular Polygon Geometry

Intuition

A regular n-gon splits into n identical isosceles triangles from its centre. That single triangle gives every measurement — the interior angle, the inscribed-circle radius, and the circumscribed-circle radius — through a cotangent or cosecant of π/n.

Definition

For a regular polygon of nn sides, each of length ss:

  • Interior angle: (n−2) 180∘n\dfrac{(n-2)\,180^\circ}{n}.
  • Inradius (inscribed circle, touching each side): r=s2cot⁡πnr = \dfrac{s}{2}\cot\dfrac{\pi}{n}.
  • Circumradius (through the vertices): R=s2csc⁡πnR = \dfrac{s}{2}\csc\dfrac{\pi}{n}.

Regular n-gon inradius

r=s2cot⁡πn,interior angle=(n−2)180∘nr = \dfrac{s}{2}\cot\dfrac{\pi}{n}, \qquad \text{interior angle} = \dfrac{(n-2)180^\circ}{n}

Worked example

Find the interior angle of a regular hexagon.
  1. Interior angle =(n−2) 180∘n= \dfrac{(n-2)\,180^\circ}{n} with n=6n = 6.
  2. =4⋅180∘6=720∘6= \dfrac{4\cdot 180^\circ}{6} = \dfrac{720^\circ}{6}.
Answer:120∘120^\circ.

From the bank · past-year question

Example 2Properties of TriangleHARD
What is the diameter of a circle inscribed in a regular polygon of 12 sides, each of length 1 cm?

[Q12 · Sep · 2022]

Drill 1 more on regular polygon geometry

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Incircle, Circumcircle & the Central Angle

    Inradius, circumradius, central angle

    r=Δs,R=abc4Δ,∠BOC=2 ∠BACr = \dfrac{\Delta}{s}, \quad R = \dfrac{abc}{4\Delta}, \quad \angle BOC = 2\,\angle BAC
  • Regular Polygon Geometry

    Regular n-gon inradius

    r=s2cot⁡πn,interior angle=(n−2)180∘nr = \dfrac{s}{2}\cot\dfrac{\pi}{n}, \qquad \text{interior angle} = \dfrac{(n-2)180^\circ}{n}

Mastery check — 4 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Properties of TriangleHARD
Consider the following for the items that follow: In a triangle PQRPQR, PP is the largest angle and cos⁡P=13\cos P=\frac{1}{3}. Further the in-circle of the triangle touches the sides PQPQ, QRQR and RPRP at NN, LL and MM respectively such that the lengths PNPN, QLQL and RMRM are nn, n+2n+2, n+4n+4 respectively where nn is an integer.
What is the length of the smallest side?

[Q46 · Apr · 2023]

Example 2Properties of TriangleEASY
What is the interior angle of a regular octagon of side length 2 cm?

[Q26 · Apr · 2021]

Example 3Properties of TriangleHARD
Consider the following for the items that follow: In a triangle PQRPQR, PP is the largest angle and cos⁡P=13\cos P=\frac{1}{3}. Further the in-circle of the triangle touches the sides PQPQ, QRQR and RPRP at NN, LL and MM respectively such that the lengths PNPN, QLQL and RMRM are nn, n+2n+2, n+4n+4 respectively where nn is an integer.
What is the value of nn?

[Q45 · Apr · 2023]

Example 4Properties of TriangleEASY
What is the length of the chord of a unit circle which subtends an angle θ\theta at the centre ?

[Q44 · Sep · 2019]

Drill every past-year question on this subtopic

6 questions from the bank — paginated, with cart and Word-export support.

Drill the 6 questions