This mini-lesson covers the Geometry & Measures strand of Eduqas GCSE Maths: angles (lines, points, parallel lines & triangles), polygons, Pythagoras, trigonometry (SOHCAHTOA), area & volume, bearings, and the Higher-tier topics of circle theorems and vectors.
The right-angled triangle is at the heart of this strand: Pythagoras and trigonometry both live here.
Work through each screen, answer the questions as you go (some are wordy, most are calculations) and collect ⭐ stars. Press Start when you're ready.
Angle facts
Angles on lines, at points & in triangles
These four facts unlock most angle questions. Learn them cold:
Angles on a straight line add to 180°.
Angles around a point add to 360°.
Angles in a triangle add to 180°.
Angles in a quadrilateral add to 360°.
a + b + c = 180°the three interior angles of any triangle
Worked example
A triangle has angles 70° and 55°. Find the third angle.
180 − 70 − 55 = 55°
Common slip: don't confuse "on a line" (180°) with "at a point" (360°). A quick sketch shows which one applies.
Calculate
Your turn — angles on a line
1Two angles sit on a straight line. One is 115°. Work out the other angle, in degrees.
°
Hint: angles on a straight line add to 180°, so 180 − 115.
Parallel lines
Angles in parallel lines
When a straight line (a transversal) crosses two parallel lines, three pairs of angles appear:
Corresponding angles (F-shape) are equal.
Alternate angles (Z-shape) are equal.
Co-interior angles (C-shape) add to 180°.
Alternate angles between the parallels are equal — the classic Z-shape.
Give a reason: exams award marks for the name of the fact. Always write "alternate angles", "corresponding angles" or "co-interior angles" alongside your working.
Polygons
Interior & exterior angles
For a polygon with n sides:
interior angle sum = (n − 2) × 180°exterior angles always add to 360°
For a regular polygon every angle is equal, so:
each exterior angle = 360° ÷ n
each interior angle = 180° − exterior angle
Worked example
Find each interior angle of a regular octagon (n = 8).
Exterior angle = 360 ÷ 8 = 45°
Interior angle = 180 − 45 = 135°
Shortcut: interior + exterior always make a straight line (180°), because they sit next to each other at a vertex.
Calculate
Your turn — interior angle sum
2Work out the sum of the interior angles of a pentagon (5 sides), in degrees.
°
Hint: (n − 2) × 180 with n = 5, so 3 × 180.
Calculate
Your turn — exterior angle
3Work out the size of each exterior angle of a regular hexagon (6 sides), in degrees.
°
Hint: exterior angles add to 360°, so 360 ÷ 6.
Quick check
Angles in a triangle
?What do the three interior angles of any triangle add up to?
Pythagoras
Pythagoras' theorem
In a right-angled triangle, the two shorter sides (legs) and the longest side (the hypotenuse, opposite the right angle) are linked by:
a² + b² = c²c is the hypotenuse — always the longest side
The famous 3-4-5 triangle: 3² + 4² = 5². To find a leg instead, rearrange to c² − a².
Finding a shorter side? Subtract, don't add: b² = c² − a². Adding gives an answer larger than the hypotenuse, which is impossible.
Calculate
Your turn — find the hypotenuse
4A right-angled triangle has legs of length 6 cm and 8 cm. Work out the length of the hypotenuse, in cm.
cm
Hint: 6² + 8² = 36 + 64 = 100, then take the square root.
Trigonometry
SOHCAHTOA
In a right-angled triangle, label the sides relative to the angle: opposite (facing it), adjacent (next to it) and hypotenuse (longest). Then:
sin = O/H · cos = A/H · tan = O/AS-O-H C-A-H T-O-A
Pick the ratio that uses the two sides in your problem, then solve.
Worked example
Angle 30°, hypotenuse 10. Find the opposite side.
Use SOH: sin30 = opp ÷ 10 → opp = 10 × sin30
sin30° = 0.5, so opp = 10 × 0.5 = 5
Calculate
Your turn — find the opposite side
5A right-angled triangle has an angle of 30° and a hypotenuse of 10 cm. Work out the length of the side opposite the 30° angle, in cm.
cm
Hint: SOH → opp = hyp × sin30° = 10 × 0.5.
Quick check
Which ratio is sine?
?Which of these correctly defines sin of an angle in a right-angled triangle?
Match game
Shape ⇄ formula
Tap a shape or rule on the left, then its matching formula on the right.
Area
Areas you must know
Keep these formulae to hand — they come up in almost every paper:
rectangle = l × wtriangle = ½ × base × height · circle = πr²
Worked example
Find the area of a triangle with base 8 cm and height 5 cm.
Area = ½ × 8 × 5 = ½ × 40 = 20 cm²
Watch the height: in a triangle the "height" is the perpendicular height (straight up from the base), not the slanted side.
Calculate
Your turn — area of a triangle
6A triangle has base 8 cm and perpendicular height 5 cm. Work out its area, in cm².
cm²
Hint: ½ × base × height = ½ × 8 × 5.
Volume
Volume of prisms
A prism has the same cross-section all the way through. Its volume is:
volume = area of cross-section × lengthcuboid = l × w × h · cylinder = πr² × h
A cuboid is a prism: multiply its three dimensions together.
Units: area is in cm², volume in cm³. If lengths are in different units, convert first.
Calculate
Your turn — volume of a cuboid
7A cuboid measures 2 cm by 3 cm by 4 cm. Work out its volume, in cm³.
cm³
Hint: l × w × h = 2 × 3 × 4.
Bearings
Bearings
A bearing gives direction as a three-figure angle measured clockwise from North:
Always three figures — write 060°, not 60°.
Measured clockwise from the North line.
East = 090°, South = 180°, West = 270°.
A bearing of 060° points 60° clockwise of due North.
Back bearings: to reverse a direction, add 180° (or subtract 180° if the result would exceed 360°).
Circle theorems · Higher
Circle theorems (Higher tier)
A handful of circle theorems solve most higher-tier circle problems:
The angle in a semicircle is 90°.
The angle at the centre is twice the angle at the circumference (same arc).
Angles in the same segment are equal.
Opposite angles of a cyclic quadrilateral add to 180°.
angle at centre = 2 × angle at circumferenceboth angles must stand on the same arc
Higher only: circle theorems are assessed on the Higher tier of Eduqas GCSE Maths, not Foundation.
Calculate · Higher
Your turn — angle at the centre
8The angle at the circumference, standing on an arc, is 40°. Work out the angle at the centre standing on the same arc, in degrees.
°
Hint: angle at centre = 2 × angle at circumference = 2 × 40.
Transformations & vectors · Higher
Transformations & vectors
The four transformations move or resize a shape: translation (slide), reflection (flip), rotation (turn) and enlargement (scale).
A column vector describes a translation — top number is movement right, bottom number is movement up:
translate by ( 3 , −2 )3 right and 2 down; negatives reverse the direction
Higher only: vector arithmetic and vector geometry proofs sit on the Higher tier. To add vectors, add the top numbers together and the bottom numbers together.
Sort it
Acute or obtuse?
An acute angle is less than 90°; an obtuse angle is between 90° and 180°. Tap an angle, then tap the box it belongs in.
◹ Acute (<90°)
◺ Obtuse (>90°)
Recap
The whole of Geometry & Measures
Angles: on a line = 180°, at a point = 360°, in a triangle = 180°; parallel-line facts (corresponding, alternate, co-interior).
Polygons: interior sum = (n − 2) × 180°; exterior angles add to 360°.
Pythagoras: a² + b² = c² in a right-angled triangle.
Trigonometry: SOHCAHTOA — sin = O/H, cos = A/H, tan = O/A.
Area & volume: triangle = ½bh, circle = πr²; prism = cross-section × length.
Bearings: three figures, clockwise from North.
Higher only: circle theorems, transformations & vectors.
You've covered every Geometry & Measures sub-topic in Eduqas GCSE Maths — Foundation content plus the Higher-tier circle theorems and vectors. Press Finish to see your score.
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