This mini-lesson covers the core Geometry & Measures content of CCEA GCSE Mathematics: angles, polygons, Pythagoras, trigonometry (SOHCAHTOA), area & volume, circle theorems, transformations, bearings and constructions.
The right-angled triangle sits at the heart of Pythagoras and trigonometry.
Honest heads-up: these quick questions test your recall of facts, formulae and methods — they can't mark full written working. In the real exam you must always show each step.
Work through each screen, answer the questions as you go and collect ⭐ stars. Press Start when you're ready.
Geometry · Angle facts
Angle facts
The key angle facts you must recall:
straight line = 180° · full turn = 360°angles in a triangle add to 180° · angles in a quadrilateral add to 360°
The two angles sit on a straight line, so ? = 180 − 55 = 125°.
Look for the setup: "on a straight line" → subtract from 180. "Around a point" → subtract from 360. Always name the fact you use.
Calculate
Your turn — angles on a line
1Two angles sit on a straight line. One is 125°. Work out the other angle, in degrees.
°
Hint: angles on a straight line add to 180°, so 180 − 125.
Geometry · Polygons
Angles in polygons
For a polygon with n sides:
interior angle sum = (n − 2) × 180°exterior angles always add to 360° · exterior = 360 ÷ n for a regular polygon
Two routes, same answer: the interior angle sum of a hexagon is (6−2)×180 = 720°, and 720 ÷ 6 = 120° for a regular one. The exterior route is quicker.
Calculate
Your turn — exterior angle
2A regular polygon has 8 sides (a regular octagon). Work out the size of one exterior angle, in degrees.
°
Hint: exterior angles add to 360°, so 360 ÷ 8.
Geometry · Pythagoras
Pythagoras' theorem
In a right-angled triangle, the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides:
a² + b² = c²c is the hypotenuse — always opposite the right angle
6² + 8² = 36 + 64 = 100, so c = √100 = 10.
Finding a shorter side? Rearrange: a² = c² − b². Subtract, don't add, when the missing side is not the hypotenuse.
Calculate
Your turn — find the hypotenuse
3A right-angled triangle has the two shorter sides 5 cm and 12 cm. Work out the length of the hypotenuse, in cm.
cm
Hint: 5² + 12² = 25 + 144 = 169, then √169.
Geometry · Trigonometry
Trigonometry — SOHCAHTOA
In a right-angled triangle, the three ratios link an angle to two sides:
SOH · CAH · TOAsin = Opp/Hyp · cos = Adj/Hyp · tan = Opp/Adj
Worked example
Find the opposite side when the angle is 30° and the hypotenuse is 10 cm.
Use sin: sin 30° = Opp ÷ 10
Opp = 10 × sin 30° = 10 × 0.5 = 5 cm
Label first: the side opposite the angle is "Opp", the side next to it (not the hypotenuse) is "Adj". Pick the ratio that uses the two sides you know or want.
Quick check
Choose the ratio
?You know the opposite and adjacent sides and want the angle. Which ratio do you use?
Measures · Area
Area formulae
The area formulae you must know by heart:
rectangle = b × h · triangle = ½ × b × hcircle area = πr² · circle circumference = 2πr (or πd)
Worked example — triangle
A triangle has base 10 cm and height 6 cm.
Area = ½ × 10 × 6 = ½ × 60 = 30 cm²
Watch the height: for a triangle, height is the perpendicular height (straight up from the base), not a slanted side.
Calculate
Your turn — area of a triangle
4A triangle has base 8 cm and perpendicular height 5 cm. Work out its area, in cm².
cm²
Hint: ½ × 8 × 5 = ½ × 40.
Measures · Volume
Volume of a prism
A prism has the same cross-section all the way through. Its volume is:
volume = area of cross-section × lengthcuboid = length × width × height · cylinder = πr² × height
Worked example — cuboid
A cuboid measures 2 cm × 3 cm × 4 cm.
Volume = 2 × 3 × 4 = 24 cm³
Units: volume is always in cubic units (cm³, m³). If you get an answer in cm², you've found an area by mistake.
Calculate
Your turn — volume of a cuboid
5A cuboid measures 5 cm × 3 cm × 4 cm. Work out its volume, in cm³.
cm³
Hint: multiply all three: 5 × 3 × 4.
Match game
Shape ⇄ formula
Tap a shape on the left, then its matching area or volume formula on the right.
Geometry · Circle theorems
Circle theorems
Key circle theorems to recall:
The angle in a semicircle is 90° (angle subtended by a diameter).
The angle at the centre is twice the angle at the circumference (same arc).
A tangent meets a radius at 90°.
Any triangle drawn from the ends of a diameter to the circle has a right angle at the circumference.
Always quote the theorem in the exam — the reason earns marks, not just the number.
Quick check
Angle in a semicircle
?A triangle is drawn from the two ends of a diameter to a point on the circle. What is the angle at that point?
Geometry · Transformations
Transformations
Four transformations you must describe fully:
Translation — slide by a vector (right/up). Shape and size unchanged.
Reflection — flip across a mirror line. Give the line's equation.
Rotation — turn about a centre by an angle and direction.
Enlargement — scale by a factor from a centre. Size changes.
Which change size? Translation, reflection and rotation are congruent (same size). Only enlargement changes the size — by the scale factor.
Quick check
Which transformation changes size?
?Which single transformation can change the size of a shape?
Measures · Bearings
Bearings
A bearing is an angle measured:
clockwise · from North · 3 figuresalways written with three digits, e.g. 060°, 135°, 270°
Start at North, turn clockwise to the direction, and read off the angle.
Three figures always: a bearing of "sixty degrees" is written 060°, not 60°. Due East is 090°, due South is 180°, due West is 270°.
Calculate
Your turn — a bearing
6A ship sails due West. Write its bearing as a number of degrees (measured clockwise from North).
°
Hint: North = 000, East = 090, South = 180, so West = ...
Geometry · Constructions
Constructions & loci
Constructions use a pair of compasses and a straight edge only — no protractor for the angle. Key ones:
Perpendicular bisector — the set of points equidistant from two ends of a line.
Angle bisector — the set of points equidistant from two lines.
A locus is the path of all points obeying a rule (e.g. a fixed distance from a point is a circle).
Leave your arcs showing: the compass arcs are the evidence of a correct construction — don't rub them out.
Quick check
Points equidistant from two ends
?Which construction gives all the points that are the same distance from both ends of a line segment?
Sort it
Same size or changes size?
Sort each transformation or fact: does it keep a shape the same size (congruent), or can it change the size? Tap an item, then the box it belongs in.
🟰 Same size
🔍 Changes size
Recap
The whole of Geometry & Measures
Angles: straight line 180°, point 360°, triangle 180°, quadrilateral 360°.
Polygons: interior sum = (n−2)×180°; exterior angles add to 360°.
Pythagoras: a² + b² = c² for right-angled triangles; subtract to find a shorter side.
Trigonometry: SOH CAH TOA links an angle to two sides.
Area & volume: rectangle b×h, triangle ½b×h, circle πr²; prism = cross-section × length.
Circle theorems: angle in a semicircle = 90°; centre angle = twice circumference angle.