Angle Sum of a Polygon & Clock Angles

📑 On this page

⚡ The short answer

A polygon with n sides has interior angles adding to (n − 2) × 180°; its exterior angles always total 360°. The angle between clock hands is |30H − 5.5M| (over 180°? take 360 minus it). A pie-chart sector is its share of 360°: 1% = 3.6°. The trap: forgetting the hour hand moves.

🎬 The 30-second story

The quarter-past-three bet

“Quarter past three. Both hands on the 3. Zero degrees. Bet you a cold coffee.”

🕒

Aarav

Library break, staring at the wall clock.

“The hour hand doesn't sit on the 3 waiting for you. It's been creeping for 15 minutes.”

⏳

Zoya

“Half a degree a minute: 7.5° past the 3. You owe her a coffee!”

🧮

Meher

Tara's maths: The hour hand moves 0.5° every minute. At 3:15 it sits at 90 + 7.5 = 97.5° from the 12, while the minute hand is at 90°. Gap = 7.5°, never zero.

What angle rules do you actually need for CLAT?

Almost every angle fact comes from two numbers: a straight line is 180° and a full turn is 360°. Open a laptop flat and screen plus keyboard make a straight line; spin once on your chair and you've turned 360°.

🧩 At a glance

The six rules that solve almost everything

➖Straight line = 180°

Angles on one side of a line add to 180°: 65° leaves 115°.

🔄Round a point = 360°

Angles meeting at one point, like pizza slices, add to 360°.

✖️Vertically opposite are equal

Two crossing lines: angles facing each other across the X are equal.

🛤️Parallel lines: equal pairs

A road cutting parallel tracks: corresponding (F) and alternate (Z) angles are equal.

🧩Co-interior add to 180°

The two angles between the tracks on one side (C-shape) add to 180°.

🔺Triangle = 180°

An exterior angle equals the two interior angles opposite it added together.

📌 Complementary vs supplementary

Complementary makes a Corner (90°); Supplementary a Straight line (180°). 35° has complement 55° and supplement 145°.

Why do a polygon's angles add up to (n − 2) × 180°?

Stand at one corner of any polygon and draw diagonals to every corner you can reach. A pentagon splits into 3 triangles, a hexagon into 4: always two fewer than the sides, because your own corner and its two neighbours give no diagonal.

🧮 The formula

Slice the polygon into triangles

🔺

Number of triangles

n − 2

📐

Angles in each triangle

180°

⬡

Sum of interior angles

(n − 2) × 180°

📐 The figure

Three polygons, three angle sums

Pentagon

3 triangles × 180°

Regular: each angle 108°

Hexagon

4 triangles × 180°

Regular: each angle 120°

Octagon

6 triangles × 180°

Regular: each angle 135°

Figures are not to scale.

The exterior angle shortcut: walk the boundary

Walk round a hexagonal running track. At each corner you turn a little: that's the exterior angle. Back at the start you face the same way, so you've turned exactly 360°, whether the track has 3 sides or 30. For a regular polygon, each exterior angle is 360 ÷ n and each interior angle is 180° minus that.

Sides (n)	Angle sum	Each interior (regular)	Each exterior (regular)
3	180°	60°	120°
4	360°	90°	90°
5	540°	108°	72°
6	720°	120°	60°
8	1,080°	135°	45°
10	1,440°	144°	36°
12	1,800°	150°	30°

→ scroll to see more

Polygon questions solved: from warm-up to CLAT level

✍️ Worked example Warm-up ⏱ 20 sec

How many sides does the badge have?

Meher's fest committee orders regular-polygon badges. Each interior angle of the badge is 156°. How many sides does it have?

1
Don't touch (n − 2) × 180. Go to the exterior angle: 180 − 156 = 24°.
2
Exterior angles add to 360°, so n = 360 ÷ 24 = 15.

Answer: 15 sides

⚡ The exam-hall shortcut

Interior given → find exterior → divide 360. One subtraction, one division, no equation.

✍️ Worked example CLAT level ⏱ 45 sec

Angles in a ratio

Kabir's new shop sign is a pentagon whose angles are in the ratio 2 : 3 : 4 : 4 : 5. What is its largest angle?

1
Angle sum of a pentagon = (5 − 2) × 180 = 540°.
2
Total parts = 2 + 3 + 4 + 4 + 5 = 18, so one part = 540 ÷ 18 = 30°.
3
Angles: 60°, 90°, 120°, 120°, 150° (total 540° ✓).

Answer: 150°

⚡ The exam-hall shortcut

Largest = 5/18 × 540 = 150. A convex shape can't have an angle of 180° or more; if your ratio gives one, recheck the sum.

✍️ Worked example Tricky ⏱ 40 sec

High-fives and diagonals

Twelve friends sit round a table. Every pair who are not neighbours high-five once. How many high-fives?

1
All pairs: 12 × 11 ÷ 2 = 66.
2
Neighbours are the 12 sides of the 12-gon, so they're left out: 66 − 12 = 54.
3
That's the diagonal formula n(n − 3)/2 = 12 × 9 ÷ 2 = 54.

Answer: 54 high-fives

⚡ The exam-hall shortcut

Each person skips themselves and two neighbours: 12 × 9 ÷ 2.

How do you find the angle between the hands of a clock?

A clock is two runners on a 360° track. Measure both hands from the 12 and subtract. Work out their speeds and the formula writes itself.

⚔️ Side by side

Minute hand vs hour hand

🏃Minute hand

360° in 60 min

•Speed 6° per minute
•At H:M it is 6M from the 12

🐢Hour hand

30° per hour

•Speed 0.5° per minute
•At H:M it is 30H + 0.5M from the 12

Test	Minute hand	Hour hand
Gap closes at	6 − 0.5 = 5.5° per minute	the hour hand never stops
Angle at H:M	\|30H + 0.5M − 6M\|	= \|30H − 5.5M\|

Over 180°? The smaller angle is 360° minus it.

📐 The figure

Two times, two angles

3:40

|90 − 220| = 130°

10:10

|300 − 55| = 245 → 360 − 245 = 115°

Figures are not to scale.

✍️ Worked example CLAT level ⏱ 20 sec

The lecture starts at 7:20

Aarav's online class starts at 7:20. What angle do the hands make then?

1
30H = 30 × 7 = 210; 5.5M = 5.5 × 20 = 110.
2
|210 − 110| = 100°, under 180°, so that's it.

Answer: 100°

⚡ The exam-hall shortcut

At 7:00 the gap is 210°. Twenty minutes close 20 × 5.5 = 110°. 210 − 110 = 100.

✍️ Worked example Tricky ⏱ 45 sec

When do the hands meet after 4?

Rohan says the hands overlap at 4:20. When, exactly, between 4 and 5 do they coincide?

1
At 4:00 the minute hand is 120° behind the hour hand.
2
It gains 5.5° a minute, so it catches up in 120 ÷ 5.5 = 240/11 = 21 9/11 minutes.
3
At 4:20 the gap is still |120 − 110| = 10°. Close, not right.

Answer: 4:21 9/11 (about 4:21:49)

⚡ The exam-hall shortcut

Hands coincide at 60H/11 minutes past H: 60 × 4 ÷ 11 = 21 9/11.

🎛️ Try it yourself — change any number

Polygon angles and clock hands

Sides of the polygon Clock: hour Clock: minutes

Sum of interior angles

1,080°

Each interior angle (regular)

135°

Clock-hand angle

100°

(n − 2) × 180; clock angle = |30H − 5.5M|

Defaults: an octagon (1,080°, 135° each) and 7:20 (100°). Type 15 sides to check the badge (156°). Then set the clock to 3:15 and settle Aarav's bet: 7.5°.

Clocks that run fast or slow

A clock that gains 2 minutes an hour shows 62 minutes per 60 real ones, so every hand runs 62/60 as fast. Find what it shows per true hour, then scale.

What is the fastest way to crack clock and polygon questions?

Event	Every	In 12 hours	In 24 hours
Hands together (0°)	65 5/11 min	11 times	22 times
Hands at right angles	about 32.7 min	22 times	44 times
Hands opposite (180°)	65 5/11 min	11 times	22 times

→ scroll to see more

Why 11, not 12? In 12 hours the minute hand laps the dial 12 times and the hour hand once, so it gains only 11 laps. Each lap brings one overlap, one straight line and two right angles.

⚠️ Three traps examiners love

1. Freezing the hour hand on the hour (Aarav's 3:15 mistake).
2. Using 'each angle = 180 − 360/n' on a polygon that isn't regular; only the sum is fixed.
3. Reporting 245° when the question means the smaller angle, 115°.

How do angles hide inside a CLAT caselet?

Since 2020, CLAT's angles turn up in pie charts more than polygons. A pie is 360°, so a sector of x° is x/360 of the total: 1% = 3.6°.

🥧 The split

College fest budget: ₹4,80,000 split by angle

Stage and sound108°
Food90°
Decor72°
Prizes54°
Publicity36°

💡 Work in degrees until the last step

Here 1° = ₹4,80,000 ÷ 360 = ₹1,333.33, a messy number. So find gaps in degrees first: stage minus prizes = 54°, which is 3/20 of ₹4,80,000 = ₹72,000. One conversion, not two.

🧪 CLAT-style practice (not a past question) Pie-chart angles

A college fest committee has a budget of ₹4,80,000. The pie chart of its plan gives each head a sector: stage and sound 108°, food 90°, decor 72°, prizes 54° and publicity 36°. A week before the fest, the committee moves ₹24,000 from food to publicity. The total budget does not change.

What is the central angle of the food sector in the revised chart?

A66°

B72°

C85°

D108°

Reveal answer + reasoning

Answer: B

Convert the change, not the sector: ₹24,000 is 1/20 of ₹4,80,000, and 1/20 of 360° = 18°. Food falls from 90° to 72°.
Check: food was ₹1,20,000, now ₹96,000; 96,000/4,80,000 × 360 = 72°. Time: 30 sec.

Why the others fail

A — Subtracts 24 as if ₹24,000 were 24°. Rupees and degrees are different units.
C — ₹24,000 is 5% of the budget, but 5% is 18°, not 5°.
D — Adds the 18° to food instead of taking it away.

How has CLAT asked angles and clocks?

📝 Asked in CLAT 2018 · Q136 Angle sum and bisectors

ABCD is a quadrilateral in which measures of angle D and angle C are 60 degree, and 100 degree respectively. If the internal bisectors of angle A and angle B meet at P, then measure of angle APB is

Find the measure of angle APB.

A80°

B90°

C100°

D110°

Reveal answer + reasoning

Answer: A

A quadrilateral's angles total 360°, so A + B = 360 − (60 + 100) = 200°.
In triangle APB the half-angles at A and B add to 100°, so ∠APB = 180 − 100 = 80°.
Shortcut: ∠APB = (C + D)/2. Time: 30 sec.

Why the others fail

B — Assumes bisectors meet at a right angle. That only happens when A + B = 180°.
C — Stops at (A + B)/2 = 100° and forgets to subtract it from 180°.
D — No valid route gives 110°; it's there for guessers.

📝 Asked in CLAT 2019 · Q49 Faulty clock

A clock gains 2 minutes every hour.

Then the angle traversed by the second hand in one minute is:

A360

B370

C390

D372

Reveal answer + reasoning

Answer: D

Our solution. In one true hour the clock shows 62 minutes, so every hand runs 62/60 as fast as normal.
A correct second hand sweeps 360° a minute; this one sweeps 360 × 62/60 = 372° in one real minute.
Reading 'one minute' as a minute on the faulty clock gives 360°, but then the gain would be pointless. Time: 25 sec.

Why the others fail

A — The angle for a correct clock: ignores the 2-minute gain.
B — Adds 2 × 5 = 10° as if the gain were in degrees.
C — Treats the gain as 2 minutes per minute-hand step; 390° is 360 × 65/60, not 62/60.

Which angle myths cost you marks?

💥 Myth vs fact

Four angle myths

🙅 Myth

At 3:15 the hands overlap.

Tap to bust it →

✅ Fact

The hour hand has moved 7.5° past the 3. The gap is 7.5°.

🙅 Myth

More sides means a bigger total of exterior angles.

Tap to bust it →

✅ Fact

The exterior angles of any polygon add to 360°: one full turn.

🙅 Myth

Each angle of any hexagon is 120°.

Tap to bust it →

✅ Fact

Only for a regular hexagon. An irregular one only promises a sum of 720°.

🙅 Myth

The hands overlap 24 times a day.

Tap to bust it →

✅ Fact

22 times. The minute hand gains just 11 laps every 12 hours.

Can you crack these five angle questions?

🎯 Quick quiz

Five angles, no protractor

0 / 5

Each exterior angle of a regular polygon-shaped tabletop is 40°.

Q1. How many sides does the tabletop have?

The hostel mess opens at 2:30.

Q2. What angle do the clock hands make at 2:30?

A straight road crosses two parallel railway tracks. The two co-interior angles it makes between the tracks are in the ratio 4 : 5.

Q3. What is the larger of these two angles?

In a pie chart of Zoya's ₹20,000 monthly spending, rent takes a 126° sector.

Q4. How much does she spend on rent?

A hostel clock loses 3 minutes every hour. It is set right at 9 AM.

Q5. What does it show at 5 PM real time?

🎯 Angles and polygons in a nutshell

Straight line 180°, full turn 360°.
Polygon sum = (n − 2) × 180°; exterior angles total 360°.
Interior angle given? Exterior first, then n = 360 ÷ exterior.
Clock angle = |30H − 5.5M|; over 180°, take 360 minus it.
Faulty clock: scale by what it shows per true hour.
Pie charts: 1% = 3.6°; work in degrees, convert once.

📎 Formula card

Lines: straight line 180°; round a point 360°; vertically opposite, corresponding and alternate angles equal; co-interior add to 180°.
Triangle: 180°; exterior angle = sum of the two opposite interior angles.
Polygon (n sides): sum = (n − 2) × 180°; exterior sum = 360°; diagonals = n(n − 3)/2.
Regular polygon: each exterior = 360/n; each interior = 180 − 360/n.
Quadrilateral: bisectors of A and B meet at (C + D)/2.
Clock: minute hand 6°/min, hour hand 0.5°/min; angle = |30H − 5.5M|; hands meet 60H/11 min past H, 11 times in 12 hours.
Pie chart: sector = share × 360°; 1% = 3.6°; 1° = 1/360 of the total.

Frequently asked questions

How do you calculate the angle between the hands of a clock?

Use |30H − 5.5M|, where H is the hour and M the minutes. 30H is where the hour hand starts; 5.5M is how far the minute hand gains on it (6° a minute against 0.5°). At 7:20 that is |210 − 110| = 100°. Over 180°? Subtract from 360° for the smaller angle.

What is the sum of the interior angles of a polygon?

It is (n − 2) × 180°, where n is the number of sides. From one corner, any polygon splits into n − 2 triangles of 180° each. A pentagon gives 540°, a hexagon 720° and an octagon 1,080°. The exterior angles of any polygon add up to 360°.

How do you find the number of sides of a regular polygon from one angle?

Go through the exterior angle. Subtract the interior angle from 180°, then divide 360 by the result. For an interior angle of 156°, the exterior angle is 24°, so the polygon has 360 ÷ 24 = 15 sides. That beats solving (n − 2) × 180 ÷ n = 156.

How many times do the hands of a clock overlap in a day?

22 times, not 24. In 12 hours the minute hand makes 12 laps and the hour hand one, so the minute hand gains 11 laps and passes the hour hand 11 times, once every 65 5/11 minutes. Right angles happen 44 times a day.

How do you convert a pie chart percentage into degrees?

Multiply the percentage by 3.6, because a whole pie is 360°. So 25% is 90° and 15% is 54°. Going back, divide degrees by 3.6: a 126° sector is 35% of the total. In CLAT caselets, find differences in degrees first and convert to rupees once.