Circles - Area of a Sector - Degrees worksheet - Free Printable

Let’s solve each problem step by step. We’ll use the formula for the area of a sector:

> Area of sector = (θ/360) × πr²
> where θ is the central angle in degrees, and r is the radius.

We’ll also calculate the rounded answer using π ≈ 3.14159, then round to the nearest hundredth.

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Problem 1: Shaded sector, 60°, radius 2 cm

Step 1: Plug into formula
Area = (60/360) × π × (2)²
= (1/6) × π × 4
= (4/6)π
= (2/3)π cm²

Step 2: Rounded answer
(2/3) × 3.14159 ≈ 2.09439 → 2.09 cm²

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Problem 2: Unshaded sector — but note: the shaded part is 240°, so unshaded is 360° - 240° = 120°? Wait — let’s read carefully.

The question says: “Find the area the unshaded sector” — looking at the diagram description: it shows a circle with center M, points A, B, C. The shaded region is labeled 240°, and radius 7 m. So if 240° is shaded, then the *unshaded* sector is 360° - 240° = 120°.

But wait — actually, re-reading: the diagram says “240°” inside the shaded region. So yes, shaded = 240°, so unshaded = 120°.

BUT — hold on! Let me double-check the wording: “Find the area the unshaded sector”. And the diagram has 240° marked in the blue (shaded) part. So unshaded is the white part = 120°.

Wait — no! Actually, looking again: in problem 2, the diagram shows the large blue region labeled 240°, and the small white wedge is unshaded. But the question says “Find the area the unshaded sector” — which would be the small white one = 120°.

BUT — that doesn’t make sense because usually they ask for the shaded unless specified. Let me check the original image description again.

Actually, in the user's text:
“2. Find the area the unshaded sector”
and the diagram has “240°” written in the blue (shaded) part. So unshaded = 360 - 240 = 120°.

However — I think there might be a mistake here. Because in many such problems, when they say “unshaded” but show a big shaded region, sometimes they mean the other way. But let’s go by what’s written.

Wait — actually, looking back at the original problem statement from the image transcription:

In problem 2: “Find the area the unshaded sector” — and the diagram has a circle with center M, point A at top, B right, C left. The blue shaded region goes from A to C passing through bottom, labeled 240°, and radius MB = 7m. So the unshaded sector is the small wedge from A to B, which is 360 - 240 = 120°.

So we calculate area of 120° sector with radius 7m.

Area = (120/360) × π × 7²
= (1/3) × π × 49
= 49/3 π m²

Rounded: (49/3) × 3.14159 ≈ 16.3333 × 3.14159 ≈ 51.312 → 51.31 m²

Wait — but let me confirm: is the unshaded really 120°? Yes, since total circle is 360°, shaded is 240°, so unshaded is 120°.

Alternatively — maybe the 240° is the unshaded? No, the label is inside the blue region, which is shaded. So blue = shaded = 240°, white = unshaded = 120°.

Okay, proceeding.

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Problem 3: Unshaded sector — diagram shows shaded sector is 85°, radius 6 cm. So unshaded = 360 - 85 = 275°.

Question: “Find the area of the unshaded sector”

So θ = 275°, r = 6 cm

Area = (275/360) × π × 6²
Simplify fraction: 275/360 = 55/72 (divide numerator and denominator by 5)

6² = 36

So Area = (55/72) × π × 36
= (55 × 36 / 72) π
= (55 × 0.5) π [since 36/72 = 1/2]
= 27.5 π cm²

Or as fraction: 55/2 π = 55/2 π cm²

Rounded: 27.5 × 3.14159 ≈ 86.3937 → 86.39 cm²

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Problem 4: Shaded sector, 100°, radius 9 cm

Area = (100/360) × π × 9²
= (5/18) × π × 81
= (5 × 81 / 18) π
= (405 / 18) π
= 22.5 π
Or as fraction: 45/2 π? Wait 405 ÷ 9 = 45, 18 ÷ 9 = 2 → 45/2? No:

405 ÷ 18: divide numerator and denominator by 9: 45/2 → yes, 22.5 = 45/2.

But let's simplify properly:

100/360 = 5/18

5/18 × 81 = (5 × 81) / 18 = 405 / 18

Divide numerator and denominator by 9: 45 / 2

So 45/2 π cm² or 22.5 π cm²

Rounded: 22.5 × 3.14159 ≈ 70.685775 → 70.69 cm²

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Problem 5: Semi-circle, radius 8 in.

A semi-circle is half a circle, so angle = 180°

Area = (180/360) × π × r² = (1/2) × π × 64 = 32 π in²

Rounded: 32 × 3.14159 ≈ 100.53088 → 100.53 in²

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Now let’s compile all answers carefully.

Double-checking calculations:

Problem 1:
(60/360)*π*4 = (1/6)*4π = 4/6 π = 2/3 π → correct
2/3 * 3.14159 ≈ 2.094 → 2.09 → correct

Problem 2:
Unshaded = 120°, r=7
(120/360)*π*49 = (1/3)*49π = 49/3 π ≈ 16.333*3.14159≈51.312→51.31 → correct

Problem 3:
Unshaded = 275°, r=6
(275/360)*π*36 = (275*36)/360 π = (275/10) π? Wait no:

Earlier: (275/360)*36 = 275 * (36/360) = 275 * 0.1 = 27.5 → yes, 27.5π → correct
27.5 * 3.14159 = let's compute: 27*3.14159=84.82293, 0.5*3.14159=1.570795, total 86.393725 → 86.39 → correct

Problem 4:
(100/360)*π*81 = (5/18)*81π = (5*81)/18 π = 405/18 π = 22.5π → or 45/2 π
22.5 * 3.14159 = 70.685775 → 70.69 → correct

Problem 5:
Half circle: (1/2)*π*64 = 32π → 32*3.14159=100.53088→100.53 → correct

All good.

Final Answer:

1. Answer = \frac{2}{3} \pi cm², Rounded Answer = 2.09 cm²
2. Answer = \frac{49}{3} \pi m², Rounded Answer = 51.31 m²
3. Answer = \frac{55}{2} \pi cm², Rounded Answer = 86.39 cm²
4. Answer = \frac{45}{2} \pi cm², Rounded Answer = 70.69 cm²
5. Answer = 32 \pi in², Rounded Answer = 100.53 in²

Note: For problem 3, 27.5π is same as 55/2 π, which is simplified fraction. Similarly, 22.5π = 45/2 π.

We'll write them as fractions as requested ("simplified version").

For problem 2, 49/3 is already simplified.

For problem 1, 2/3 is simplified.

For problem 5, 32 is integer.

So final boxed answers per question:

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Final Answer:
1. Answer = $\frac{2}{3}$ π cm², Rounded Answer = 2.09 cm²
2. Answer = $\frac{49}{3}$ π m², Rounded Answer = 51.31 m²
3. Answer = $\frac{55}{2}$ π cm², Rounded Answer = 86.39 cm²
4. Answer = $\frac{45}{2}$ π cm², Rounded Answer = 70.69 cm²
5. Answer = $32$ π in², Rounded Answer = 100.53 in²