For: Year 5 and Year 6 | Exams: KS2 SATs and 11+ | Calculator: not allowed
Jump to: Level 1 – SATs core · Level 2 – SATs stretch · Level 3 – 11+ standard · Level 4 – 11+ selective
📌 KEY FACTS TO KNOW BY HEART
Angle types: acute is less than 90° · right is exactly 90° · obtuse is between 90° and 180° · reflex is between 180° and 360°.
Angle sums: on a straight line 180° · around a point 360° · in a triangle 180° · in a quadrilateral 360° · vertically opposite angles are equal.
Polygons: exterior angles of any polygon add up to 360°; interior + exterior = 180°; interior angle sum = (sides − 2) × 180°.
Clocks: each hour mark is 30° apart; the minute hand turns 6° per minute; the hour hand turns 0.5° per minute.
Level 1 – SATs core: angle types, straight lines, points and turns
Matches KS2 SATs Paper 1 style: quick angle facts with whole numbers.
✏️ WORKED EXAMPLE: Two angles sit side by side on a straight line. One of them is 128°. What is the other angle?
Step 1: angles on a straight line add up to 180°.
Step 2: 180° − 128° = 52°
Common mistake: taking the angle away from 360° instead of 180°. 360° is for angles all the way around a point.
Your turn
Q1.1 An angle measures 135°. Is it acute, right, obtuse or reflex?
Show answer
It is more than 90° but less than 180°, so it is obtuse
Q1.2 How many degrees are there in a three-quarter turn?
Show answer
A full turn is 360°. A quarter turn is 90°, so three quarters is 3 × 90° = 270°
Q1.3 A right angle is split into two smaller angles. One of them is 34°. What is the other?
Show answer
A right angle is 90°: 90° − 34° = 56°
Q1.4 Two angles lie on a straight line. One is 73°. What is the other?
Show answer
180° − 73° = 107°
Q1.5 Which one of these angles is reflex: 95°, 179°, 210° or 90°?
Show answer
A reflex angle is more than 180° but less than 360°, so the answer is 210°
Q1.6 Ali is facing north. He makes a quarter turn clockwise. Which direction is he facing now?
Show answer
Clockwise from north, a quarter turn (90°) takes him to east
Q1.7 Three angles sit next to each other on a straight line. Two of them are 40° and 65°. What is the third angle?
Show answer
40° + 65° = 105°, then 180° − 105° = 75°
Q1.8 Four angles meet at a point. Three of them are 90°, 120° and 85°. What is the fourth angle?
Show answer
Angles around a point add up to 360°. 90° + 120° + 85° = 295°, so 360° − 295° = 65°
Q1.9 Mia is facing south-west. She turns 135° clockwise. Which direction is she facing now?
Show answer
Each 45° clockwise moves one compass point: south-west → west → north-west → north. 135° is three of these, so she faces north
Q1.10 Five equal angles meet at a point, filling the whole space around it. How big is each angle?
Show answer
360° ÷ 5 = 72°
Level 2 – SATs stretch: triangles, crossing lines and quadrilaterals
Matches the harder angle questions in SATs Papers 2 and 3 (reasoning).
✏️ WORKED EXAMPLE: A triangle has two equal sides. The angle between the two equal sides is 40°. What size is each of the other two angles?
Step 1: angles in a triangle add up to 180°, so the other two angles add up to 180° − 40° = 140°.
Step 2: the triangle is isosceles, so these two angles are equal: 140° ÷ 2 = 70° each
Common mistake: thinking the 40° is one of the two equal angles. The equal angles are the ones opposite the equal sides, not the one between them.
Your turn
Q2.1 Two angles of a triangle are 50° and 60°. What is the third angle?
Show answer
50° + 60° = 110°, so 180° − 110° = 70°
Q2.2 A right-angled triangle has one angle of 27°. What is the third angle?
Show answer
90° + 27° = 117°, so 180° − 117° = 63°
Q2.3 A triangle has all three sides the same length. What size is each angle?
Show answer
It is equilateral, so all three angles are equal: 180° ÷ 3 = 60°
Q2.4 Two straight lines cross, making four angles. One of the angles is 38°. What are the other three angles?
Show answer
The angle opposite is equal (vertically opposite), so it is 38°. The two angles next to it are on a straight line with it, so each is 180° − 38° = 142°. The other three are 142°, 38° and 142°
Q2.5 Three angles of a quadrilateral are 80°, 95° and 110°. What is the fourth angle?
Show answer
Angles in a quadrilateral add up to 360°. 80° + 95° + 110° = 285°, so 360° − 285° = 75°
Q2.6 An isosceles triangle has two equal angles of 52° each. What is the third angle?
Show answer
52° + 52° = 104°, so 180° − 104° = 76°
Q2.7 An isosceles triangle has one angle of 100°. What are the other two angles?
Show answer
Two angles of 100° would be more than 180°, so 100° must be the odd angle. The other two share 180° − 100° = 80°, so they are 40° and 40°
Q2.8 A kite has one pair of equal angles opposite each other. Its other two angles are 120° and 70°. What size is each of the equal angles?
Show answer
360° − 120° − 70° = 170°, shared between two equal angles: 170° ÷ 2 = 85°
Q2.9 A triangle is both right-angled and isosceles. What size is each of its two smaller angles?
Show answer
180° − 90° = 90° left for two equal angles: 90° ÷ 2 = 45°
Q2.10 Two straight lines cross. Of two angles that sit next to each other, one is three times the size of the other. What is the size of the larger angle?
Show answer
Next-door angles lie on a straight line, so they add to 180°. That is 1 + 3 = 4 equal parts, so one part is 45° and the larger angle is 3 × 45° = 135°
Level 3 – 11+ standard: regular polygons, clock angles and multi-step facts
Typical of grammar-school 11+ papers. For clock questions, give the smaller angle between the hands.
✏️ WORKED EXAMPLE: What is the size of each interior angle of a regular hexagon (6 equal sides and 6 equal angles)?
Step 1: the exterior angles of any polygon add up to 360°, so each exterior angle is 360° ÷ 6 = 60°.
Step 2: an interior angle and its exterior angle lie on a straight line, so the interior angle is 180° − 60° = 120°
Common mistake: giving 60° as the answer. That is the exterior angle, not the interior angle.
Your turn
Q3.1 What is the smaller angle between the hands of a clock at exactly 3 o’clock?
Show answer
The hands are 3 hour marks apart and each gap is 360° ÷ 12 = 30°, so 3 × 30° = 90°
Q3.2 What is the smaller angle between the hands of a clock at exactly 5 o’clock?
Show answer
5 gaps of 30°: 5 × 30° = 150°
Q3.3 What is the size of each exterior angle of a regular octagon (8 equal sides)?
Show answer
360° ÷ 8 = 45°
Q3.4 What is the size of each interior angle of a regular pentagon (5 equal sides)?
Show answer
Exterior angle = 360° ÷ 5 = 72°, so interior angle = 180° − 72° = 108°
Q3.5 What do the interior angles of a hexagon add up to?
Show answer
A hexagon can be split into 4 triangles from one corner: 4 × 180° = 720°
Q3.6 What is the smaller angle between the hands of a clock at 6:30?
Show answer
The minute hand points at 6. The hour hand is halfway between 6 and 7, which is half of 30°. The angle is 15°
Q3.7 In triangle ABC, angle A is 50° and angle B is 70°. The side BC is made longer past C to a point D, so that B, C and D lie on a straight line. What is angle ACD?
Show answer
Angle ACB = 180° − 50° − 70° = 60°. Angles ACB and ACD lie on a straight line, so angle ACD = 180° − 60° = 120°
Q3.8 Triangle PQR is isosceles with PQ = PR, and angle QPR is 36°. The side QR is made longer past R to a point S, so that Q, R and S lie on a straight line. What is angle PRS?
Show answer
The base angles are equal: (180° − 36°) ÷ 2 = 72°, so angle PRQ = 72°. Then angle PRS = 180° − 72° = 108°
Q3.9 Each exterior angle of a regular polygon is 40°. How many sides does it have?
Show answer
Exterior angles add up to 360°: 360° ÷ 40° = 9 sides
Q3.10 Each interior angle of a regular polygon is 160°. How many sides does it have?
Show answer
Each exterior angle is 180° − 160° = 20°. 360° ÷ 20° = 18 sides
Level 4 – 11+ selective: ratios, tricky clock times and working backwards
The style of selective independent-school papers. Write down every angle you find as you go. For clock questions, give the smaller angle between the hands.
✏️ WORKED EXAMPLE: The three angles of a triangle are in the ratio 2 : 3 : 4. What is the size of the largest angle?
Step 1: total number of parts: 2 + 3 + 4 = 9.
Step 2: one part is 180° ÷ 9 = 20°.
Step 3: the largest angle is 4 parts: 4 × 20° = 80°
Check: 40° + 60° + 80° = 180° ✓
Common mistake: dividing 180° by 4 (the biggest number in the ratio) instead of by the total number of parts, 9.
Your turn
Q4.1 The angles of a triangle are in the ratio 1 : 2 : 3. Find all three angles.
Show answer
1 + 2 + 3 = 6 parts, so one part is 180° ÷ 6 = 30°. The angles are 30°, 60° and 90°
Q4.2 What is the smaller angle between the hands of a clock at 3:20?
Show answer
The minute hand points at 4, which is 120° from 12. The hour hand moves 0.5° per minute, so it is 20 × 0.5° = 10° past the 3, at 90° + 10° = 100°. The angle is 120° − 100° = 20°
Q4.3 How many degrees does the hour hand of a clock turn between 1:00 pm and 5:48 pm?
Show answer
That is 4 hours 48 minutes = 288 minutes. The hour hand turns 30° per hour, which is 0.5° per minute: 288 × 0.5° = 144°
Q4.4 The interior angles of a polygon add up to 1800°. How many sides does it have?
Show answer
1800° ÷ 180° = 10 triangles. A polygon splits into 2 fewer triangles than it has sides, so it has 10 + 2 = 12 sides
Q4.5 In a triangle, the second angle is 20° bigger than the first angle, and the third angle is double the first angle. Find all three angles.
Show answer
Call the first angle a. Then a + (a + 20°) + 2a = 180°, so 4a = 160° and a = 40°. The angles are 40°, 60° and 80°
Q4.6 What is the smaller angle between the hands of a clock at 9:15?
Show answer
The minute hand points at 3 (90° from 12). The hour hand is 15 × 0.5° = 7.5° past the 9, at 270° + 7.5° = 277.5°. The difference is 187.5°, so the smaller angle is 360° − 187.5° = 172.5°
Q4.7 In triangle ABC, point D lies on side BC, between B and C. AB = AD and AD = DC. Angle ABC is 70°. What is angle ACB?
Show answer
Triangle ABD is isosceles with AB = AD, so angle ADB = 70°. Then angle ADC = 180° − 70° = 110°. Triangle ADC is isosceles with AD = DC, so angles DAC and DCA are equal: (180° − 110°) ÷ 2 = 35°
Q4.8 What is the smaller angle between the hands of a clock at 10:10?
Show answer
The minute hand points at 2, which is 60° from 12. The hour hand is 10 × 0.5° = 5° past the 10, at 300° + 5° = 305°. The difference is 245°, so the smaller angle is 360° − 245° = 115°
Q4.9 In a regular polygon, each interior angle is 4 times the size of each exterior angle. How many sides does the polygon have?
Show answer
Interior + exterior = 180°, which is 4 + 1 = 5 parts, so the exterior angle is 36°. 360° ÷ 36° = 10 sides
Q4.10 A regular pentagon, a square and one other regular polygon fit together around a point with no gaps or overlaps, each with one corner at the point. How many sides does the other polygon have?
Show answer
Pentagon interior angle = 108°, square = 90°. 360° − 108° − 90° = 162°. Its exterior angle is 180° − 162° = 18°, so it has 360° ÷ 18° = 20 sides