Geometry

Geometry with Diagrams

Shaded regions, circles, Pythagoras and angles — every practice question comes with its own diagram

Explanation & Worked Examples

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Shaded Regions

The trick to nearly every shaded-area question is the same: shaded = whole shape − the bits removed. Work out each piece with its own area formula, then add or subtract.

Shape	Area
Square (side x)	x²
Circle (radius r)	πr²
Quarter circle (radius r)	¼πr²
Semicircle (radius r)	½πr²

Worked Example — Four Quadrants in a Square

A square has side x. At each corner a quarter-circle of radius x/2 is drawn and shaded. Find the area of the unshaded centre.

1Square area = x²

2Four quarter-circles of radius x/2 join into one full circle: π(x/2)² = ¼πx²

3Unshaded centre = x² − ¼πx²

Watch the radius: if a circle fits exactly inside a square of side x, its radius is x/2 — not x. Using the wrong radius is the most common mistake here.

Circles: Ring, Sector & Arc

Keep answers in terms of π unless told otherwise.

Area of a circle = πr² | Circumference = 2πr

Ring (annulus) = πR² − πr² (outer minus inner)

Sector area = (θ/360) × πr² | Arc length = (θ/360) × 2πr

Worked Example — Sector

A sector has radius 6 cm and angle 60°. Find its area.

1Fraction of the circle = 60/360 = 1/6

2Area = (1/6) × π × 6² = (1/6) × 36π = 6π cm²

Don't mix them up: the sector formula uses r² (area), the arc formula uses 2r (length). Check whether the answer should be in cm² or cm.

Pythagoras' Theorem

In any right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides:

a² + b² = c²

where c is the hypotenuse (always the longest side, opposite the right angle).

Worked Example — Find the Hypotenuse

The two short sides are 6 cm and 8 cm.

1c² = 6² + 8² = 36 + 64 = 100

2c = √100 = 10 cm

Finding a short side? Rearrange: a² = c² − b². You subtract when one of the given lengths is the hypotenuse, and add when both are short sides.

Angles & Polygons

Angles on a straight line add to 180°.
Angles around a point add to 360°.
The three angles of a triangle add to 180°.

Regular Polygons

Each exterior angle = 360 ÷ n

Each interior angle = (n − 2) × 180 ÷ n (or simply 180 − exterior)

Worked Example — Interior Angle of a Hexagon

Find one interior angle of a regular hexagon (6 sides).

1Interior = (6 − 2) × 180 ÷ 6 = 4 × 180 ÷ 6

2= 720 ÷ 6 = 120°

Exterior vs interior: the exterior angle (360 ÷ n) is a popular wrong answer. They always add to 180°, so double-check which one the question wants.

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Practice Questions

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