Shaded regions with tangents and triangles

8 minAS Level1.4.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can find the angles and lengths hiding in a circle diagram (using tangents, radii and right-angled triangles), then break a shaded region into shapes you know and add or subtract.

Shaded regions are escape rooms

The examiner shades a weird shape and asks for its area or perimeter. You never need a new formula. You need to spot which known shapes make it up:

$\text{shaded} = \text{(shape you know)} \pm \text{(another shape you know)}$

Your shapes: sectors ( $\tfrac12 r^2\theta$ ), triangles ( $\tfrac12 ab\sin C$ , using two sides $a$ , $b$ and the angle $C$ between them, or $\tfrac12 \times$ base $\times$ height) and segments (sector minus triangle, from the previous lesson). The skill is finding the missing angle or length first.

Clues and what they unlock

Clue in the diagram	Fact to use
a tangent at $A$	angle $OAT = \tfrac{\pi}{2}$ (tangent $\perp$ radius)
a right angle	Pythagoras and SOH CAH TOA
two radii $OA$ , $OB$	triangle $OAB$ is isosceles
a chord of known length	split the isosceles triangle in half to make two right-angled triangles: $\sin\tfrac{\theta}{2} = \dfrac{\tfrac12 AB}{r}$
any two sides and the angle between	$\tfrac12 ab\sin C$ , cosine rule

Find angles in radians (radian mode), because they'll go straight into $r\theta$ and $\tfrac12 r^2\theta$ .

Worked example: Find the angle, then the region

A circle has centre $O$ and radius $5$ cm. A point $T$ is $13$ cm from $O$ . The tangent from $T$ touches the circle at $A$ , and $OT$ meets the circle at $B$ . Find the area and the perimeter of the region bounded by $AT$ , $TB$ and the arc $AB$ .

Have a go first?

Quick check

Multiple choiceQuestion 1 · Recall

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