Circle & Semicircle Area and Perimeter

📑 On this page

⚡ The short answer

A circle's area is πr² and its circumference 2πr. A semicircle has half the area, πr²/2, but its perimeter is πr + 2r: half the curve plus the diameter. A sector of angle θ is θ/360 of the circle. CLAT's traps: dropping that diameter, and squaring the diameter instead of the radius.

🎬 The 30-second story

The pizza paradox

“Two 8-inch pizzas or one 12-inch, same price? Two. That's 16 inches!”

🍕

Aarav

Fest night, one budget.

“Inches measure across. You eat the area.”

📏

Zoya

“Large: π × 6² ≈ 113 sq in. Two mediums: 2 × π × 4² ≈ 101. Big one wins!”

🧮

Meher

Halve each diameter, then square.

Tara's maths: Area grows with the square of the radius: 36π against 32π, so the one 12-inch pizza gives 12.5% more. Pizza sizes are diameters, so halve first; π cancels when you only compare.

Why is the area of a circle πr²?

Wrap a string round anything round, a bangle, a cup, a wheel, and divide by the distance across. You always get about 3.14. That ratio is π, so circumference = π × diameter = 2πr.

🧮 The formula

Cut a pizza into thin slices and lay them top-to-tail

↔️

Length: half the crust

½ × 2πr = πr

↕️

Height: one slice

the radius, r

⭕

Area of the circle

πr × r = πr²

Lay thin wedges alternately point-up and point-down and they make an almost-rectangle: long side half the crust (πr), short side one slice (r). Area = πr × r = πr². Thinner slices, straighter edges.

📌 Radius first, always

Every circle formula runs on the radius. If the passage gives a diameter (pizza sizes, pipe widths), halve it first. Squaring the diameter makes the area four times too big, and that number will be in the options.

How do you find the area and perimeter of a semicircle?

Cut a circle along a diameter. The area halves cleanly: πr²/2. The perimeter doesn't, because the cut creates a new straight edge: half the curve (πr) plus the diameter (2r).

📐 The figure

The three shapes, one radius

Circle

Area πr² · Circumference 2πr

Semicircle

Area πr²/2 · Perimeter πr + 2r

Sector

Area (θ/360) × πr² · Arc (θ/360) × 2πr

Figures are not to scale.

⚔️ Side by side

Full circle vs semicircle, radius 7 cm

⭕Full circle

π = 22/7, r = 7

•Area = 22/7 × 49 = 154 cm²
•Circumference = 44 cm

🌗Semicircle

same r, cut in half

•Area = 154 ÷ 2 = 77 cm²
•Perimeter = 22 + 14 = 36 cm

Test	Full circle	Semicircle
Area	πr²	πr²/2, exactly half
Boundary	2πr	πr + 2r, more than half

⚠️ Half the circumference is not the perimeter

For r = 7, half the circumference is 22 cm; the perimeter is 36 cm. Fencing a semicircular garden needs the diameter too. Only when the straight side is shared with a wall or another shape may the passage want the curve alone.

How do you find the area of a sector, a quadrant or a ring?

A sector is a pizza slice: two radii and the arc between them. It's a fraction of the circle, and the fraction is the angle out of 360. A quadrant (90°) is a quarter; a semicircle (180°) is a half.

🥧 The split

A 72° sector is one-fifth of the circle

Sector (72°)72°
Rest of the circle288°

Rings: the path around a pond

A ring (a running track, a path round a pond) is a big circle minus a small one: π(R² − r²) = π(R + r)(R − r). Given only the path's width and not the pond's radius? The area cannot be determined.

Wheels: one turn, one circumference

A rolling wheel moves forward one circumference per turn: distance = revolutions × 2πr. Watch for a cm-to-km change hiding in it.

Circle questions solved: from warm-up to CLAT level

✍️ Worked example Warm-up ⏱ 40 sec

How many turns to college?

Meher's cycle wheel has a radius of 35 cm. College is 1.1 km away. How many full turns does the wheel make? (Take π = 22/7.)

1
35 is a multiple of 7, so use 22/7: circumference = 2 × 22/7 × 35 = 220 cm.
2
1.1 km = 1,10,000 cm. Turns = 1,10,000 ÷ 220 = 500.

Answer: 500 revolutions

⚡ The exam-hall shortcut

Convert the wheel instead: 220 cm = 2.2 m, and 1,100 ÷ 2.2 = 500.

✍️ Worked example CLAT level ⏱ 75 sec

Fencing and turfing a semicircular garden

Kabir's café has a semicircular garden with a straight edge of 28 m. Fencing the whole boundary costs ₹50/m and turfing ₹20/m². Total bill? (π = 22/7.)

1
Straight edge = diameter, so r = 14 m.
2
Perimeter = 22/7 × 14 + 28 = 44 + 28 = 72 m; fencing = 72 × 50 = ₹3,600.
3
Area = ½ × 22/7 × 196 = 308 m²; turf = 308 × 20 = ₹6,160.
4
Total = ₹9,760.

Answer: Fencing ₹3,600 + turf ₹6,160 = ₹9,760

⚡ The exam-hall shortcut

Semicircle perimeter = 36/7 × r, so r = 14 gives 72 in one line. Learn 36/7 like 22/7.

✍️ Worked example Tricky ⏱ 60 sec

Resurfacing a running track

A circular ground has a radius of 70 m. A 7 m wide track runs round its outside. Resurfacing costs ₹10/m². Total cost? (π = 22/7.)

1
Outer radius R = 70 + 7 = 77 m; inner r = 70 m.
2
Area = π(R + r)(R − r) = 22/7 × 147 × 7 = 22 × 147 = 3,234 m².
3
Cost = 3,234 × 10 = ₹32,340.

Answer: ₹32,340 (track area 3,234 m²)

⚡ The exam-hall shortcut

Never square 77. (R + r)(R − r) gives 147 × 7, and the width 7 cancels the 7 in 22/7.

What is the fastest way to do circle sums in the exam?

Radius	Area πr²	Circumference 2πr	Semicircle edge πr + 2r
3.5	38.5	22	18
7	154	44	36
10.5	346.5	66	54
14	616	88	72
21	1,386	132	108

→ scroll to see more

With π = 22/7, the circumference is 44 for every 7 of radius. Otherwise use 3.14; the gap (about 0.04%) never changes the right option. If options are written in terms of π, keep π as a letter.

🎛️ Try it yourself — change any number

Circle and semicircle in one go

Radius of the garden (m)

Area

615.75

Circumference

87.96

Semicircle area

307.88

Semicircle perimeter

71.98

Area = πr²; Circumference = 2πr; semicircle perimeter = πr + 2r (don't forget the diameter)

Exact π shows 615.75 and 87.96; 22/7 gives 616 and 88. Now type 28: the area jumps four times (about 2,463), the circumference only doubles.

Estimate first. A circle's area is a bit more than 3r² and its circumference a bit more than 6r. For r = 10, expect just over 300 and 60. An option of 31.4 or 3,140 has slipped a decimal.

How do circles hide inside a CLAT Quant caselet?

CLAT won't ask 'find the area of a circle of radius 7'. It describes a park or a plot, mixes shapes, and fires several questions off one figure. Break it into pieces you know, then add or subtract.

📊 The data

A society park, piece by piece (area in m²)

Rectangular lawn · 40 m × 28 m

1,120

Semicircular flower bed · diameter 28 m

308

Fountain (cut out) · radius 3.5 m

38.5

Green area left · 1,120 + 308 − 38.5

1,389.5

Add the bed (it sticks out); subtract the fountain (it sits inside).

🧪 CLAT-style practice (not a past question) Semicircle in a composite figure

A housing society's park is a rectangle 40 m long and 28 m wide. A semicircular flower bed is built outward on one of the 28 m sides, so that side becomes the bed's diameter. A circular fountain of radius 3.5 m sits inside the rectangle. The society will fence the outer boundary of the park, including the curved edge of the flower bed, at ₹25 per metre. (Take π = 22/7.)

What will the fencing cost?

A₹3,400

B₹3,800

C₹4,500

D₹4,900

Reveal answer + reasoning

Answer: B

The fence runs round the outside: two 40 m sides, the far 28 m side and the bed's curve. The side where the bed attaches is now inside the park. Arc = 22/7 × 14 = 44 m. Boundary = 40 + 40 + 28 + 44 = 152 m; cost = 152 × 25 = ₹3,800. The fountain is a decoy. About 60 seconds.

Why the others fail

A — Rectangle only (136 m); forgets the curve.
C — Also fences the 28 m side now inside the park (180 m).
D — Uses the full circumference, 88 m (196 m in all).

How did CLAT 2020 ask about a semicircle?

CLAT 2020 built five questions (Q146–150) on one semicircular plot. The words survive, but the figure is just four letters on the page. It shows a semicircle on diameter AC, centre O, and B at the top of the arc, straight above O. The pole is at O, so the rope BO is a radius.

📐 The figure

The CLAT 2020 plot, redrawn

Semicircular plot, centre O

Area πr²/2 · Perimeter πr + 2r

B is the top of the arc. OA = OB = OC = r.

Not to scale. AC is the diameter; the goat is tied at O.

📝 Asked in CLAT 2020 · Q146 Area equals perimeter

Direction: In the figure given below, a goat is tied to a pole (at point O) which is the center of the semi-circular plot. The area of the plot is equal to its perimeter. BO is the length of the rope through which the goat is tied to the pole. Answer to the closest decimal. [Figure, described: a semicircle with diameter AC along the bottom and centre O on it; B is the top of the arc, directly above O.]

The length of the rope is

A2 + π

B2 + π/2

C2 + 2/π

Reveal answer + reasoning

Answer: C

The method. Rope BO = radius r. Semicircle area = πr²/2; perimeter = πr + 2r (arc plus diameter AC). Set equal and divide by r: πr/2 = π + 2, so r = 2 + 4/π ≈ 3.27.
The snag. That isn't an option. The official key says C, 2 + 2/π ≈ 2.64, which solves πr²/2 = πr + r: the arc plus only one radius, not the whole diameter. We show the key's answer, but correct semicircle maths gives 3.27.
In the hall. If your clean answer isn't there, re-check once, then move on: a blank costs nothing, a wrong answer 0.25. About 75 seconds.

Why the others fail

A — 2 + π ≈ 5.14 is the (π + 2) multiplier from the perimeter, not a radius.
B — 2 + π/2 ≈ 3.57 comes from no version of the equation.
D — Dropping the diameter: πr²/2 = πr gives r = 2. The classic slip.

📝 Asked in CLAT 2020 · Q147 Area of the plot

Same figure and data as Q146: a semi-circular plot, centre O, whose area is equal to its perimeter. Answer to the closest decimal.

The area of the plot is

A18.13

B21.83

C28.13

D11.83

Reveal answer + reasoning

Answer: B

With the key's r = 2 + 2/π ≈ 2.637: πr² = 4π + 8 + 4/π ≈ 12.57 + 8 + 1.27 = 21.84, option B.
But πr² is a full circle. The semicircular plot with that radius is ≈ 10.92, and with the correct radius (≈ 3.27) it is ≈ 16.83, equal to its perimeter as promised. Neither is offered, so B works only if you take the key's radius and don't halve. About 45 seconds.

Why the others fail

A — Matches no reading of the figure.
C — Matches no reading; above every possible value.
D — Near the key-radius semicircle (10.92) but not equal to it.

The lesson. The tested idea is the semicircle's perimeter. Write πr + 2r every time. Q148–150 on triangle ABC belong to the triangles page.

Which circle myths cost you marks?

💥 Myth vs fact

Four circle myths

🙅 Myth

A semicircle's perimeter is half the circumference.

Tap to bust it →

✅ Fact

Add the diameter: πr + 2r. For r = 7, 36 cm, not 22.

🙅 Myth

Double the radius, double the area.

Tap to bust it →

✅ Fact

Area goes up 4 times (2²); the circumference only doubles.

🙅 Myth

π is exactly 22/7.

Tap to bust it →

✅ Fact

22/7 = 3.1428…; π = 3.14159…. It's an approximation, handy when r is a multiple of 7.

🙅 Myth

A ring's area is π(R − r)².

Tap to bust it →

✅ Fact

It's π(R + r)(R − r): 3,234 m² for a 7 m track round a 70 m ground, not 154 m².

Can you crack these five circle questions?

🎯 Quick quiz

Five circles, one radius each

0 / 5

A circular rangoli has a circumference of 88 cm.

Q1. What is its area? (Take π = 22/7.)

A semicircular protractor has a diameter of 14 cm.

Q2. How long is its full boundary? (Take π = 22/7.)

A café makes each round table's radius 10% longer.

Q3. By what percent does each tabletop's area grow?

A round cake of radius 21 cm is cut into three equal slices.

Q4. What is the top area of one slice? (Take π = 22/7.)

A cycle wheel of radius 42 cm turns 250 times.

Q5. How far has the cycle travelled? (Take π = 22/7.)

🎯 Circles in a nutshell

Radius first: halve any diameter.
Circle: area πr², circumference 2πr.
Semicircle: πr²/2 and πr + 2r. Never drop the diameter.
Sector: θ/360 of the circle, plus two radii on its edge.
Ring: π(R + r)(R − r). Wheel: turns × 2πr.
Radius × k means area × k².

📎 Formula card

Circle: A = πr²; C = 2πr = πd.
Semicircle: A = πr²/2; P = πr + 2r ≈ 36r/7.
Quadrant: A = πr²/4; P = πr/2 + 2r.
Sector θ: A = (θ/360)πr²; arc = (θ/360)2πr; P = arc + 2r.
Ring: π(R + r)(R − r). Wheel: distance = turns × 2πr.
π: 22/7 if r is a multiple of 7, else 3.14. r = 7 → 154, 44; r = 14 → 616, 88.
Area = boundary: circle at r = 2; semicircle at r = 2 + 4/π ≈ 3.27.

Frequently asked questions

What is the formula for the perimeter of a semicircle?

It is πr + 2r: the curved half of the circumference plus the straight diameter. With π = 22/7 that is 36/7 × r, roughly 5.14r. For a radius of 7 cm the perimeter is 22 + 14 = 36 cm. Leaving out the 2r is the commonest semicircle mistake.

How do you find the area of a semicircle?

Halve the full circle: πr²/2. A diameter of 28 m gives a radius of 14 m, a full circle of 22/7 × 196 = 616 m² and a semicircle of 308 m². Halve the diameter before squaring, or your answer comes out four times too big.

When should I use 22/7 and when 3.14 for π?

Use 22/7 when the radius or diameter is a multiple of 7, because the 7 cancels and the numbers stay whole. Use 3.14 otherwise. Both approximate π (3.14159…) closely enough never to change the right CLAT option. If the options contain π, keep it as a symbol.

How do you find the area of a sector of a circle?

A sector is a fraction of the circle: its angle over 360. Area = (θ/360) × πr² and arc = (θ/360) × 2πr. A 90° sector is a quarter, 60° a sixth. Its perimeter is the arc plus two radii, so a 45° slice of radius 7 has an edge of 5.5 + 14 = 19.5.