Sector of a circle with a central angle of 82° and radius 12 cm.

Let’s solve this step by step.

We are given a sector of a circle — that’s like a “slice” of a pie. The radius is 12 cm, and the angle at the center is 82°. We need to find the area of the shaded part — which is the segment (the curved part between the chord and the arc).

To find the area of the segment, we do:

> Area of segment = Area of sector – Area of triangle

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Step 1: Find the area of the sector

The formula for the area of a sector is:

> Area_sector = (θ / 360) × π × r²

Where:
- θ = central angle in degrees → 82°
- r = radius → 12 cm
- π ≈ 3.1416

So plug in the numbers:

Area_sector = (82 / 360) × π × 12²
= (82 / 360) × π × 144
First, calculate 82 ÷ 360 ≈ 0.227777...
Then multiply by 144: 0.227777... × 144 ≈ 32.8
Now multiply by π: 32.8 × 3.1416 ≈ ?

Let me compute more accurately:

82/360 = 41/180
So: (41/180) × π × 144
= (41 × 144 × π) / 180
Simplify 144/180 = 4/5? Wait, let's divide numerator and denominator by 36:

144 ÷ 36 = 4
180 ÷ 36 = 5
Wait, no — better to simplify 144/180 = 16/20 = 4/5? Actually:

144 ÷ 12 = 12
180 ÷ 12 = 15 → so 12/15 = 4/5? No, 144/180 = 0.8 exactly? Let’s check:

144 ÷ 180 = 0.8 → yes! Because 180 × 0.8 = 144.

So: (41 × 0.8) × π = 32.8 × π

Now 32.8 × 3.1416 ≈ let’s calculate:

32.8 × 3 = 98.4
32.8 × 0.1416 ≈ 32.8 × 0.14 = 4.592, plus 32.8 × 0.0016 ≈ 0.05248 → total ≈ 4.64448
So total ≈ 98.4 + 4.64448 ≈ 103.04448 cm²

So area of sector ≈ 103.04 cm² (we’ll keep more digits for accuracy later)

Actually, let’s use exact fractions or calculator-style precision.

Better approach:

Use π = 3.1415926535...

Area_sector = (82/360) * π * 144
= (82 * 144 * π) / 360

Calculate 82 * 144:

80*144 = 11520
2*144 = 288
Total = 11520 + 288 = 11808

Now divide by 360: 11808 ÷ 360

Divide numerator and denominator by 24:

11808 ÷ 24 = 492
360 ÷ 24 = 15 → wait, maybe easier:

11808 ÷ 360 = 11808 ÷ 36 ÷ 10 = (11808 ÷ 36) ÷ 10

11808 ÷ 36:

36 × 328 = 36×300=10800, 36×28=1008 → 10800+1008=11808 → yes!

So 11808 ÷ 36 = 328 → then ÷10 = 32.8

So again: 32.8 × π

Now 32.8 × π = 32.8 × 3.1415926535 ≈ ?

Let me compute:

32.8 × 3.1415926535

Break it down:

30 × 3.1415926535 = 94.247779605
2.8 × 3.1415926535 = ?

2 × 3.1415926535 = 6.283185307
0.8 × 3.1415926535 = 2.5132741228
Sum: 6.283185307 + 2.5132741228 = 8.7964594298

Now add to 94.247779605:
94.247779605 + 8.7964594298 = 103.0442390348

So area of sector ≈ 103.044 cm²

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Step 2: Find the area of the triangle

The triangle is formed by two radii and the chord. It’s an isosceles triangle with two sides = 12 cm, and included angle = 82°.

Formula for area of triangle when you know two sides and the included angle:

> Area_triangle = (1/2) × a × b × sin(C)

Here, a = b = 12 cm, C = 82°

So:

Area_triangle = (1/2) × 12 × 12 × sin(82°)

= 72 × sin(82°)

Now, what is sin(82°)? Use calculator:

sin(82°) ≈ 0.9902680687

So:

72 × 0.9902680687 ≈ ?

70 × 0.9902680687 = 69.318764809
2 × 0.9902680687 = 1.9805361374
Sum ≈ 71.2993009464

So area of triangle ≈ 71.299 cm²

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Step 3: Subtract to get area of segment

Area_segment = Area_sector - Area_triangle
≈ 103.044 - 71.299 = 31.745 cm²

Let’s round appropriately. Since inputs were whole numbers (82°, 12 cm), probably expect answer to 1 or 2 decimal places.

But let’s check our calculations once more for accuracy.

Recalculate sector area:

(82/360)*π*144 = (82*144*π)/360

As before, 82*144 = 11808
11808 / 360 = 32.8
32.8 * π = 32.8 * 3.141592653589793 = let’s use precise value:

32.8 * 3.141592653589793 =

Do 328 * 3.141592653589793 / 10

328 * 3.141592653589793:

300 * 3.141592653589793 = 942.4777960769379
28 * 3.141592653589793 = 87.9645943005142
Sum = 942.4777960769379 + 87.9645943005142 = 1030.4423903774521
Divide by 10 → 103.04423903774521 → same as before.

Triangle: 72 * sin(82°)

sin(82°) = sin(82) = using calculator: 0.9902680687415704

72 * 0.9902680687415704 = 71.29930094939307

Subtract:

103.04423903774521 - 71.29930094939307 = 31.74493808835214

So approximately 31.74 cm²

Depending on rounding, sometimes they want 1 decimal place.

But let’s see — if we use π = 3.14 and sin(82°) ≈ 0.99, what do we get?

Sector: (82/360)*3.14*144 = (0.227777...)*3.14*144

0.227777 * 144 = 32.8 (same)
32.8 * 3.14 = 103.0 (exactly? 32.8 * 3.14 = 32.8*3 + 32.8*0.14 = 98.4 + 4.592 = 102.992)

Triangle: 72 * 0.99 = 71.28

Difference: 102.992 - 71.28 = 31.712 → still about 31.7

But since we used more precise values, 31.74 is better.

In many textbooks, they might expect the answer rounded to 1 decimal place.

But let’s check: perhaps we should report as 31.7 cm²? Or 31.74?

Looking back at the problem — it says “12 cm” and “82°”, both integers. Probably acceptable to give answer to 1 or 2 decimals.

I think 31.7 cm² is fine, but let’s see the exact value is ~31.74, so maybe 31.7 if rounding to one decimal.

Wait — actually, in geometry problems like this, often they expect two decimal places.

Alternatively, let’s write it as 31.74 cm².

But to be safe, let’s confirm with another method.

Another way: use formula for segment area directly:

Area_segment = (r²/2) * (θ_rad - sin(θ_deg))

But θ must be in radians for that formula.

Convert 82° to radians: 82 * π / 180 = 82π/180 = 41π/90 radians

Then:

Area_segment = (1/2) * r² * (θ_rad - sin(θ_deg))
= (1/2) * 144 * (41π/90 - sin(82°))
= 72 * (41π/90 - sin(82°))

Compute 41π/90:

π ≈ 3.1415926535 → 41 * 3.1415926535 / 90

41 * 3.1415926535 = 128.8052987935
Divide by 90: 1.431169986594444...

sin(82°) ≈ 0.9902680687

So difference: 1.431169986594444 - 0.9902680687 ≈ 0.440901917894444

Multiply by 72: 72 * 0.440901917894444 ≈ ?

70 * 0.440901917894444 = 30.86313425261108
2 * 0.440901917894444 = 0.881803835788888
Sum ≈ 31.7449380884

Same result: 31.7449... cm²

So, final answer is approximately 31.74 cm²

If we round to nearest hundredth, it’s 31.74 cm².

Sometimes problems expect to use π = 3.14 and sin(82°) = 0.99, but since we did precise calculation, I’ll go with 31.74.

But let me check if the problem expects integer? Unlikely.

Perhaps 31.7 cm²? But 31.74 is more accurate.

I think for school level, they might accept 31.7 or 31.74.

Looking at similar problems, often they keep two decimals.

So I’ll go with 31.74 cm²

But wait — let’s see if we can write it as fraction? Probably not necessary.

Final decision: 31.74 cm²

But to match typical textbook answers, perhaps round to one decimal: 31.7 cm²?

No, because 31.74 is closer to 31.7 than 31.8? 31.74 is 31.7 when rounded to one decimal? No:

Rounding 31.74 to one decimal: look at second decimal is 4 < 5, so 31.7

But actually, 31.74 is already to two decimals.

I think it’s fine to leave as 31.74.

Alternatively, the problem might expect the answer as 31.7 cm².

But let’s calculate numerically once more with standard calculator values.

Assume:

π = 3.1416
sin(82°) = 0.9903

Sector: (82/360)*3.1416*144 = (0.2277778)*3.1416*144

First, 0.2277778 * 144 = 32.8 (exactly? 82/360 * 144 = 82 * 0.4 = 32.8 yes)

32.8 * 3.1416 = 32.8 * 3.1416

30*3.1416=94.248
2.8*3.1416=8.79648
Sum=103.04448

Triangle: 72 * 0.9903 = 71.3016

Difference: 103.04448 - 71.3016 = 31.74288 ≈ 31.74 cm²

Yes.

So final answer is 31.74 cm²

But perhaps the problem expects units and rounding.

I think it's safe to box 31.74

However, in some contexts, they might want it as 31.7, but I'll go with precise.

Wait — let me see if there's a mistake.

The shaded region is the segment, which is sector minus triangle — yes.

And we calculated correctly.

So:

Final Answer:
31.74 cm²