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AQA GCSE Maths (8300) · Geometry & Measures
Mini-Lesson

Geometry & Measures

This mini-lesson covers the Geometry & measures strand of AQA GCSE Maths (8300): angles, polygons, Pythagoras, trigonometry (SOHCAHTOA), area & volume, circle theorems, transformations, vectors, and bearings.

a² + b² = c²Pythagoras' theorem for a right-angled triangle (c = hypotenuse)

These games test recall of facts, formulae and methods — not full multi-step working. Answer the questions as you go and collect ⭐ stars. Press Start when you're ready.

Angle facts

Angle facts

Learn the key angle sums — they're the building blocks of every angle problem:

on a line = 180° · at a point = 360°angles in a triangle = 180°  ·  angles in a quadrilateral = 360°
Worked example

A triangle has angles 90° and 35°. Find the third.

Angles in a triangle sum to 180°.

180 − 90 − 35 = 55°

Parallel lines: co-interior (allied) angles add to 180°; alternate ("Z") and corresponding ("F") angles are equal.

Calculate

Your turn — angles in a triangle

1A triangle has two angles of 65° and 40°. Work out the third angle.
°
Hint: 180 − 65 − 40.
Polygons

Angles in polygons

For a polygon with n sides:

interior angle sum = (n − 2) × 180°exterior angles always add to 360°
Worked example

Sum of interior angles of a hexagon (n = 6).

(6 − 2) × 180 = 4 × 180 = 720°

Regular polygons: each exterior angle = 360 ÷ n. For a regular hexagon that's 60°, so each interior angle is 180 − 60 = 120°.

Calculate

Your turn — exterior angle

2Work out the size of each exterior angle of a regular pentagon (5 sides).
°
Hint: exterior angles sum to 360°, so 360 ÷ 5.
Pythagoras' theorem

Pythagoras' theorem

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

a² + b² = c²c is the hypotenuse — the longest side, opposite the right angle
3 4 5
3² + 4² = 9 + 16 = 25 = 5². The classic 3-4-5 triangle.

Finding a shorter side? Rearrange: a² = c² − b². Subtract, don't add, when the hypotenuse is already known.

Calculate

Your turn — Pythagoras

3A right-angled triangle has shorter sides 6 cm and 8 cm. Work out the length of the hypotenuse.
cm
Hint: c² = 6² + 8² = 36 + 64 = 100, then square root.
Trigonometry · SOHCAHTOA

Trigonometry: SOHCAHTOA

In a right-angled triangle, the sides are named relative to an angle: opposite, adjacent and hypotenuse.

SOH · CAH · TOAsin = O/H  ·  cos = A/H  ·  tan = O/A
Worked example

Find the opposite side when the angle is 30° and the hypotenuse is 10 cm.

sin 30° = O ÷ 10, and sin 30° = 0.5

O = 10 × 0.5 = 5 cm

Exact values to know: sin 30° = ½, cos 60° = ½, tan 45° = 1, sin 90° = 1.

Quick check

Choose the ratio

?You know the opposite and adjacent sides and want the angle. Which ratio do you use?
Area & volume

Area, circles & volume

Learn the standard formulae. For a circle of radius r:

area = πr² · circumference = πdtriangle = ½ × base × height  ·  cuboid volume = l × w × h
Worked example

Area of a circle with radius 5 cm (use π ≈ 3.14).

area = π × 5² = π × 25

≈ 3.14 × 25 = 78.5 cm²

Don't mix them up: circumference uses the diameter (πd) or 2πr; area uses r squared (πr²).

Calculate

Your turn — area of a triangle

4A triangle has base 12 cm and perpendicular height 5 cm. Work out its area.
cm²
Hint: area = ½ × base × height = ½ × 12 × 5.
Calculate

Your turn — volume of a cuboid

5A cuboid measures 4 cm by 3 cm by 5 cm. Work out its volume.
cm³
Hint: volume = length × width × height = 4 × 3 × 5.
Circle theorems

Circle theorems

The key circle theorems each give an angle relationship:

  • The angle at the centre is twice the angle at the circumference (same arc).
  • The angle in a semicircle is 90°.
  • Angles in the same segment are equal.
  • Opposite angles of a cyclic quadrilateral add to 180°.
Worked example

The angle at the circumference is 40°. Find the angle at the centre on the same arc.

Centre angle = 2 × circumference angle = 2 × 40 = 80°

Tangent facts: a tangent meets a radius at 90°, and two tangents from a point are equal in length.

Calculate

Your turn — angle at the centre

6The angle at the circumference standing on an arc is 55°. Work out the angle at the centre on the same arc.
°
Hint: the centre angle is twice the circumference angle: 2 × 55.
Bearings

Bearings

A bearing is an angle measured clockwise from North, always written with three figures.

measured clockwise from North · 3 figuresdue East = 090° · due South = 180° · due West = 270°
Worked example

The bearing of B from A is 070°. Find the bearing of A from B (the back-bearing).

Add 180°: 70 + 180 = 250°

Three figures always: a bearing of 40° must be written 040°. Back-bearing: add 180° if under 180, subtract 180° if over.

Calculate

Your turn — back-bearing

7The bearing of Q from P is 110°. Work out the bearing of P from Q.
°
Hint: 110 is less than 180, so add 180°.
Quick check

Describe the transformation

?Which transformation is described by a column vector such as (3, −2)?
Match game

Match: measure ⇄ formula

Tap a quantity on the left, then its correct formula on the right.

Sort it

Congruent or similar?

A congruent transformation keeps the shape the same size (translation, rotation, reflection). A similar transformation (enlargement, scale factor ≠ 1) changes the size but keeps the shape. Tap a transformation, then the box it belongs in.

🟰 Congruent (same size)

🔍 Similar (size changes)

Recap

The whole Geometry & Measures strand

Angles: line = 180°, point = 360°, triangle = 180°; parallel-line rules.

Polygons: interior sum = (n − 2)×180°; exterior sum = 360°.

Pythagoras: a² + b² = c² for right-angled triangles.

Trig: SOHCAHTOA — sin = O/H, cos = A/H, tan = O/A.

Area & volume: circle area = πr², triangle = ½bh, cuboid = lwh.

Circle theorems: centre = 2× circumference; semicircle = 90°.

Bearings: clockwise from North, 3 figures; back-bearing ± 180°.

Transformations & vectors: translations, rotations, reflections keep size; enlargements change it.

You've covered every Geometry & Measures sub-topic in AQA 8300. Press Finish to see your score.

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