Surface Area of Cones Worksheets - Free Printable

Let’s solve each cone surface area problem step by step.

We are told to use π = 3.14 and round answers to two decimal places.

The formula for the surface area of a cone is:

> Surface Area = π × r × l + π × r²
> or factored: SA = πr(l + r)

Where:
- r = radius of the base
- l = slant height (the diagonal side from tip to edge of base)
- We do NOT use the vertical height unless we need to find slant height using Pythagoras — but in these problems, slant height is usually given.

Wait — looking at the diagrams, some show “slant height” labeled as the long side, and others may show vertical height. Let’s check each one carefully.

Actually, in most of these, the number written along the slanted side is the slant height (l), and the number across the bottom circle is the diameter, so we must divide by 2 to get radius.

But wait — let’s look again. In diagram 1: it says “5.2 ft” on the slant, and “2.8 ft” across the base → that’s diameter? Or radius?

Looking at standard worksheet conventions: when a line goes across the circular base with a label, it’s usually the diameter. But sometimes it’s radius. Let’s check diagram 2: it shows “12.75 m” across the base — if that were radius, the cone would be huge compared to slant height 10m — which doesn’t make sense because slant height must be longer than radius. So likely, the base measurement is diameter.

Wait — actually, in diagram 2: slant height is 10 m, and base is labeled 12.75 m — if that’s diameter, radius is 6.375 m — then slant height (10) > radius (6.375) — okay, possible.

But in diagram 1: slant = 5.2 ft, base = 2.8 ft — if base is diameter, radius = 1.4 ft — then slant (5.2) > radius (1.4) — fine.

In diagram 3: slant = 24 in, base = 9 in — if base is diameter, radius = 4.5 in — slant 24 > 4.5 — okay.

So I think all base measurements are diameters. So we’ll divide by 2 to get radius.

BUT — wait! Look at diagram 4: it says “radius 5 cm” explicitly! Oh no — inconsistency!

Diagram 4: has “5 cm” with an arrow pointing to the radius — so that’s radius.

Diagram 5: “12 yd” across base — probably diameter? But also shows vertical height 10 yd — and slant height not labeled? Wait, no — in diagram 5, the slant height is not labeled — only vertical height 10 yd and base 12 yd (probably diameter). So we may need to calculate slant height using Pythagoras.

This is getting messy. Let me go one by one.

Actually, let’s re-express: For each cone, identify:

- Is the base measurement radius or diameter?
- Is the other measurement slant height or vertical height?

From visual clues:

→ If there’s a right angle symbol inside the cone (between vertical height and radius), then the third side (hypotenuse) is slant height, and we can compute it if needed.

Let’s go problem by problem.

---

Problem 1:

Cone lying on its side. Labeled: slant side = 5.2 ft, base diameter = 2.8 ft? Or radius? The line across the base is labeled 2.8 ft — and since it spans the whole circle, likely diameter. Also, no right angle shown — so probably slant height is given directly.

Assume:
- Slant height l = 5.2 ft
- Diameter = 2.8 ft → radius r = 1.4 ft

SA = πr(l + r) = 3.14 × 1.4 × (5.2 + 1.4) = 3.14 × 1.4 × 6.6

Calculate step by step:

1.4 × 6.6 = 9.24

3.14 × 9.24 = ?

3 × 9.24 = 27.72

0.14 × 9.24 = let's see: 0.1×9.24=0.924, 0.04×9.24=0.3696 → total 1.2936

So total SA = 27.72 + 1.2936 = 29.0136 ≈ 29.01 ft²

Wait — but let me use calculator-style:

3.14 × 1.4 = 4.396

4.396 × 6.6 = ?

4 × 6.6 = 26.4

0.396 × 6.6 = 0.396×6 = 2.376, 0.396×0.6=0.2376 → total 2.6136

So 26.4 + 2.6136 = 29.0136 → yes, 29.01

Okay.

---

Problem 2:

Upright cone. Vertical height = 10 m, base diameter = 12.75 m? Label says “12.75 m” across base — likely diameter. And slant height is not labeled — but there’s a right triangle inside: height 10 m, radius = half of 12.75 = 6.375 m, so slant height l = √(10² + 6.375²)

Compute:

10² = 100

6.375² = ? 6.375 × 6.375

6×6=36, 6×0.375=2.25, doubled is 4.5? Better:

6.375² = (6 + 0.375)^2 = 36 + 2×6×0.375 + (0.375)^2 = 36 + 4.5 + 0.140625 = 40.640625

So l = √(100 + 40.640625) = √140.640625 ≈ ?

What’s square root of 140.64? 11.86^2 = ? 11.8^2=139.24, 11.9^2=141.61 → so between 11.8 and 11.9

Try 11.86: 11.86² = (12 - 0.14)^2 = 144 - 2×12×0.14 + (0.14)^2 = 144 - 3.36 + 0.0196 = 140.6596 — close to 140.640625

Slightly less — try 11.859: approximately 11.86 is fine.

Actually, let’s compute exactly:

√140.640625 — note that 140.640625 = 140640625 / 1000000 — but perhaps it’s a perfect square?

Note: 6.375 = 51/8, since 6.375 = 6 + 3/8 = 51/8

So radius r = 51/8 m

Height h = 10 = 80/8 m

Then l = √( (51/8)^2 + (80/8)^2 ) = (1/8) √(51² + 80²)

51² = 2601

80² = 6400

Sum = 9001

√9001 — what is that? 94.87? 95²=9025, 94²=8836, 94.8²=8987.04, 94.9²=9006.01 — too big, 94.87²=?

Actually, 94.87² = (95 - 0.13)^2 = 9025 - 2×95×0.13 + 0.0169 = 9025 - 24.7 + 0.0169 = 9000.3169 — close to 9001

94.88² = 94.87² + 2×94.87×0.01 + 0.0001 ≈ 9000.3169 + 1.8974 + 0.0001 = 9002.2144 — too big

So √9001 ≈ 94.873

Thus l = 94.873 / 8 ≈ 11.859 m

Now SA = πr(l + r) = 3.14 × (51/8) × (11.859 + 51/8)

First, 51/8 = 6.375

l + r = 11.859 + 6.375 = 18.234

Then SA = 3.14 × 6.375 × 18.234

First, 6.375 × 18.234

Compute 6 × 18.234 = 109.404

0.375 × 18.234 = (3/8) × 18.234 = (18.234 × 3)/8 = 54.702 / 8 = 6.83775

Total = 109.404 + 6.83775 = 116.24175

Now × 3.14 = 116.24175 × 3.14

116.24175 × 3 = 348.72525

116.24175 × 0.14 = 16.273845

Total = 348.72525 + 16.273845 = 364.999095 ≈ 365.00 m²

That seems high, but let's verify.

Alternative: since we have exact fractions.

r = 51/8, l = √( (51/8)^2 + 10^2 ) = √(2601/64 + 6400/64) = √(9001/64) = √9001 / 8

SA = π r (l + r) = π * (51/8) * ( √9001 / 8 + 51/8 ) = π * (51/8) * ( (√9001 + 51)/8 ) = π * 51 * (√9001 + 51) / 64

But this is messy. Since we got approximately 365.00, and it's very close to integer, perhaps it's intended to be exact.

Notice that 51-80-94.87... but 51^2 + 80^2 = 2601 + 6400 = 9001, and 9001 is not a perfect square, so we have to use decimal.

But in our calculation, we got 364.999, which rounds to 365.00.

Perhaps the problem expects us to use the values as given without overcomplicating.

Another thought: maybe "12.75 m" is the radius? But that would make radius 12.75, height 10, then slant = √(12.75^2 + 10^2) = √(162.5625 + 100) = √262.5625 = 16.204, then SA = 3.14*12.75*(16.204 + 12.75) = large number, about 3.14*12.75*28.954 ≈ 3.14*369.1635 ≈ 1159, which is way bigger, and unlikely.

So I think diameter is correct, and SA ≈ 365.00 m².

But let's double-check calculation:

r = 12.75 / 2 = 6.375 m

h = 10 m

l = sqrt(6.375^2 + 10^2) = sqrt(40.640625 + 100) = sqrt(140.640625) = 11.859 (as before)

l + r = 11.859 + 6.375 = 18.234

πr = 3.14 * 6.375 = let's calculate: 3.14 * 6 = 18.84, 3.14 * 0.375 = 1.1775, total 20.0175

Then 20.0175 * 18.234 = ?

20 * 18.234 = 364.68

0.0175 * 18.234 ≈ 0.319095

Total 364.999095 — yes, 365.00 when rounded.

Okay.

---

Problem 3:

Cone with slant height 24 in, base diameter 9 in? Label "9 in" across base — likely diameter.

So r = 4.5 in, l = 24 in

SA = πr(l + r) = 3.14 * 4.5 * (24 + 4.5) = 3.14 * 4.5 * 28.5

First, 4.5 * 28.5 = 4.5 * 28 + 4.5 * 0.5 = 126 + 2.25 = 128.25

Then 3.14 * 128.25 = 3 * 128.25 = 384.75, 0.14 * 128.25 = 17.955, total 402.705 ≈ 402.71 in²

---

Problem 4:

Explicitly says "radius 5 cm", and slant height 13 cm (labeled on slant side).

So r = 5 cm, l = 13 cm

SA = πr(l + r) = 3.14 * 5 * (13 + 5) = 3.14 * 5 * 18 = 3.14 * 90 = 282.60 cm²

Exactly.

---

Problem 5:

Upright cone. Base diameter 12 yd (across base), vertical height 10 yd (with right angle symbol).

So r = 6 yd, h = 10 yd

Slant height l = √(r² + h²) = √(36 + 100) = √136 = 2√34 ≈ 11.6619 yd

SA = πr(l + r) = 3.14 * 6 * (11.6619 + 6) = 3.14 * 6 * 17.6619

First, 6 * 17.6619 = 105.9714

Then 3.14 * 105.9714 = 3 * 105.9714 = 317.9142, 0.14 * 105.9714 ≈ 14.835996, total ≈ 332.750196 ≈ 332.75 yd²

---

Problem 6:

Cone on its side. Slant height 23 ft, base diameter 12 ft? Label "12 ft" across base.

So r = 6 ft, l = 23 ft

SA = 3.14 * 6 * (23 + 6) = 3.14 * 6 * 29 = 3.14 * 174 = ?

3 * 174 = 522, 0.14 * 174 = 24.36, total 546.36 ft²

---

Problem 7:

Upright cone. Base diameter 11 in, vertical height 15 in (right angle shown).

So r = 5.5 in, h = 15 in

l = √(5.5² + 15²) = √(30.25 + 225) = √255.25 ≈ 15.976 in (since 16^2=256, so √255.25≈15.976)

SA = 3.14 * 5.5 * (15.976 + 5.5) = 3.14 * 5.5 * 21.476

First, 5.5 * 21.476 = 5 * 21.476 = 107.38, 0.5 * 21.476 = 10.738, total 118.118

Then 3.14 * 118.118 = 3 * 118.118 = 354.354, 0.14 * 118.118 ≈ 16.53652, total ≈ 370.89052 ≈ 370.89 in²

---

Problem 8:

Cone on its side. Slant height 22 mm, base diameter 8 mm? Label "8 mm" across base.

So r = 4 mm, l = 22 mm

SA = 3.14 * 4 * (22 + 4) = 3.14 * 4 * 26 = 3.14 * 104 = 326.56 mm²

---

Problem 9:

Upright cone. Base diameter 16 cm, vertical height 12 cm (right angle shown).

So r = 8 cm, h = 12 cm

l = √(8² + 12²) = √(64 + 144) = √208 = 4√13 ≈ 14.4222 cm

SA = 3.14 * 8 * (14.4222 + 8) = 3.14 * 8 * 22.4222

First, 8 * 22.4222 = 179.3776

Then 3.14 * 179.3776 = 3 * 179.3776 = 538.1328, 0.14 * 179.3776 ≈ 25.112864, total ≈ 563.245664 ≈ 563.25 cm²

---

Now, let's list all answers:

1) 29.01 ft²

2) 365.00 m²

3) 402.71 in²

4) 282.60 cm²

5) 332.75 yd²

6) 546.36 ft²

7) 370.89 in²

8) 326.56 mm²

9) 563.25 cm²

I think that's all.

But let me quickly verify problem 2 again — is it possible that "12.75 m" is the radius? Because if it were, then r=12.75, h=10, l=√(12.75^2 + 10^2)=√(162.5625+100)=√262.5625=16.204, then SA=3.14*12.75*(16.204+12.75)=3.14*12.75*28.954

12.75*28.954≈ let's say 12.75*29=369.75, minus 12.75*0.046≈0.5865, so ≈369.1635, times 3.14≈1159.17, which is not nice, and the problem likely intends diameter.

In diagram 2, the label "12.75 m" is placed similarly to others where it's diameter, and in context, 365 is a clean answer, so I'll stick with that.

Similarly, for problem 5, we have r=6, h=10, l=√136≈11.6619, SA=3.14*6*17.6619≈332.75, which is fine.

Okay, final answers:

Final Answer:
1) 29.01
2) 365.00
3) 402.71
4) 282.60
5) 332.75
6) 546.36
7) 370.89
8) 326.56
9) 563.25