Chapter 2 · Kerala SSLC Class 10 Maths
Circles
Two ways to learn — a crisp formula reference, or a story that makes the theorems feel obvious.
Three real-world scenes that build each theorem from scratch.
How this works
Each scene gives you a real-world situation that naturally leads to a circle theorem. Follow the story, then open the maths box to see the formal result.
The Ferris Wheel
Two friends, Anu and Bibin, are standing on the ground watching a Ferris wheel. Two cabins, A and B, are at the top of the wheel right now. Anu and Bibin are standing at different spots on the ground, both watching the same two cabins.
Do you think we're seeing the same angle between the cabins, even though we're standing in different places?
They mark their positions — both on the ground, on the same side of the line joining A and B — and measure the angle each of them sees.
Anu's angle: ∠APB = 42°
Bibin's angle: ∠AQB = 42°
∠APB = ∠AQB ✓
Exactly equal. As long as Anu and Bibin are both on the same arc relative to AB, they will always see the same angle — no matter how far apart they stand.
Now imagine the operator, standing at the very centre of the wheel, O. The angle the operator sees between the same two cabins, ∠AOB, is always exactly double what Anu and Bibin see.
∠AOB = 2 × ∠APB
= 2 × 42°
= 84°
Now a third friend, Chinnu, walks around to the opposite side of the wheel — she is now on the other arc relative to cabins A and B. What angle does she see?
Anu's angle (near arc): ∠APB = 42°
Chinnu's angle (far arc): ∠ACB = 180° − 42°
= 138°
Chinnu doesn't see the same angle as Anu — she sees its supplement. Whenever two people stand on opposite arcs relative to the same chord, their angles always add up to 180°.
So same side → equal angles. Opposite sides → angles that add to 180°. And the wheel operator's angle tells us something else too — what fraction of the whole wheel sits between the cabins?
Operator's angle (central) = 84°
Fraction of the wheel between A and B = 84°/360°
= 7/30
The Rectangular Frame
You have a rectangular photograph — 8 cm × 6 cm. You want to frame it in a circular frame that fits perfectly so all four corners of the photo just touch the circle. Is that possible?
A rectangle inside a circle... do all four corners really land on the circle?
Yes — a rectangle always fits perfectly inside a circle, with all four corners touching it. So the rectangle is a cyclic quadrilateral — a quadrilateral with all 4 vertices on the circle.
Now look at the angles. In a rectangle, every angle is 90°. Pick opposite corners: ∠A = 90° and ∠C = 90°. Add them up:
∠A + ∠C = 90° + 90°
= 180°
The same holds for ∠B + ∠D. And here is the beautiful generalisation — this is not just about rectangles. For any cyclic quadrilateral, opposite angles always add to 180°.
It works because both opposite angles are inscribed in the same circle and together they subtend the full 360° around the centre — each pair sees exactly half.
One more useful trick: if you extend side BC of a cyclic quadrilateral ABCD outward to a point E, the exterior angle ∠DCE turns out to equal the interior angle at the opposite vertex, ∠A.
∠BCD + ∠A = 180° (cyclic quad)
∠DCE + ∠BCD = 180° (straight line)
∴ ∠DCE = ∠A
The Semicircle Surprise
Draw a circle. Mark a diameter AB — the full width across the centre. Now pick any point P on the circle (not at A or B). Connect PA and PB to form a triangle APB.
What is the angle at P? Is it always the same, no matter where P is on the circle?
Try it for three different positions of P — left, top, and right (but not on the diameter itself). Measure ∠APB each time.
P near left → ∠APB = 90°
P at top → ∠APB = 90°
P near right → ∠APB = 90°
Always 90°. Every single time. The angle at P is always a right angle. This stunning result is called Thales' theorem.
The Big Picture
Every result in this chapter connects back to one idea: the central angle is always twice the angle in the alternate arc. Same-segment angles, the semicircle rule, and cyclic quadrilaterals all follow from it.
Angle at circumference vs centre?
∠AOB = 2 × ∠APB
Two points, same arc?
∠APB = ∠AQB
Two points, opposite arcs?
∠APB + ∠AQB = 180°
Angle in a semicircle?
∠APB = 90°
What fraction of the circle is an arc?
central angle / 360°
Opposite angles of cyclic quad?
∠A + ∠C = 180°
Exterior angle of cyclic quad?
ext∠ = opposite interior ∠
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