How to Find the Surface Area of a Cone - Free Printable

Let’s solve this step by step.

We are given:
- Radius (r) = 9 inches
- Height (h) = 16 inches
- We need to find the surface area (SA) of the cone.
- The formula is: SA = πrs + πr²

But wait — we don’t have “s” (the slant height) directly. In the example, they used s = 16 in, but that’s actually the vertical height (h), not the slant height. That’s a mistake!

In a right circular cone, the slant height (s) is found using the Pythagorean theorem:

s = √(r² + h²)

So let’s calculate the correct slant height first.

Step 1: Find slant height (s)
r = 9, h = 16
s = √(9² + 16²) = √(81 + 256) = √337 ≈ 18.3576 inches

Step 2: Use the surface area formula
SA = πrs + πr²
= π * 9 * 18.3576 + π * 9²
= π * 165.2184 + π * 81
= π * (165.2184 + 81)
= π * 246.2184

Now use π ≈ 3.14
SA ≈ 3.14 * 246.2184 ≈ 773.125776

Rounded to one decimal place: 773.1 in²

Wait — but in the image’s example, they incorrectly used h instead of s and got 706.9 in². Since the problem says “solve the problem accurately,” we must use the correct method.

However… looking again at the image: in the green cone diagram, it labels the side as “16in” — which might be intended as the slant height (s), not the vertical height (h). But in the text box below, it says “h = 16in”. This is conflicting.

Let’s check both interpretations.

Interpretation A (as written in text): h = 16in → then s = √(9²+16²) = √337 ≈ 18.36 → SA ≈ 773.1 in²

Interpretation B (as labeled on diagram): if “16in” is meant to be slant height (s), then we can plug directly into formula:

SA = πrs + πr² = π*9*16 + π*81 = π*(144 + 81) = π*225 = 3.14 * 225 = 706.5 in² → which rounds to 706.5 or 706.9 depending on rounding.

The example in the image uses 706.9, which comes from:

π*144 = 3.14*144 = 452.16
π*81 = 3.14*81 = 254.34
Sum = 452.16 + 254.34 = 706.5 → but they wrote 706.9? Let me recalculate with more precision.

Actually, if you use π = 3.1416:

π*144 = 3.1416 * 144 = 452.3904
π*81 = 3.1416 * 81 = 254.4696
Sum = 706.86 → which rounds to 706.9

Ah! So they used π ≈ 3.1416 implicitly, even though they wrote π = 3.14.

But here’s the key: in the diagram, the 16in is drawn along the slanted side — so it should be ‘s’, the slant height. And in the text, they mistakenly called it ‘h’. That’s an error in labeling.

Since the diagram shows 16in on the slant edge, and the calculation matches when treating it as ‘s’, I think the intention was for s = 16in.

Therefore, to match the example’s intent (even if mislabeled), we’ll proceed with s = 16in.

Final Calculation:

SA = πrs + πr²
= π * 9 * 16 + π * 9²
= π * 144 + π * 81
= π * (144 + 81)
= π * 225

Using π = 3.1416 (to get their result):

3.1416 * 225 = 706.86 → rounded to one decimal: 706.9 in²

If we strictly follow what’s written (“h = 16in”), then we must calculate s correctly and get ~773.1 in².

But since the example in the image arrives at 706.9 by using 16 as ‘s’ (even though they called it ‘h’), and the diagram labels the slant side as 16in, I believe the intended value for ‘s’ is 16in.

To resolve this fairly: the problem says “solve the problem accurately.” Accuracy means using correct geometry. If h=16 and r=9, then s must be √(81+256)=√337≈18.36, and SA≈773.1.

But perhaps the image has a typo, and “h=16in” should be “s=16in”.

Given that the example computes SA = π(9)(16) + π(9)^2, they are clearly using 16 as ‘s’.

I think for educational purposes, we should point out the error but also show what the image intended.

However, the instruction is: “Solve the problem accurately.”

Accurate solution requires using the correct slant height if only radius and vertical height are given.

But in this case, the diagram provides the slant height visually as 16in.

Let me re-read the user’s request: “You are an educational assistant helping a student solve homework problems. The user uploaded an image containing a task. Solve the problem accurately.”

The task in the image is to compute SA using the given values. The given values in the example box say: r=9in, h=16in, and they compute using s=16in — which is wrong.

But the diagram next to it labels the slant side as 16in.

This is ambiguous.

Best approach: assume that the 16in in the diagram is the slant height (since it's drawn on the slant), and the text mistakenly called it 'h'. Because otherwise, the example doesn't make sense.

In many textbooks, when they draw a cone and label the side length, it's the slant height.

So I will go with s = 16in, r = 9in.

Then:

SA = πrs + πr² = π*9*16 + π*81 = π*(144 + 81) = π*225

Using π = 3.14 as stated in the example:

3.14 * 225 = 706.5

But the example says 706.9, which suggests they used a more precise π.

Let me calculate with π = 3.1415926535:

225 * π = 225 * 3.1415926535 = 706.858347...

Rounded to one decimal: 706.9 in²

Yes, that matches.

So even though they wrote π = 3.14, they likely used a better approximation in calculation.

For accuracy, we should use the standard value.

But to match the example's answer, we'll use the same method.

Final decision: The problem intends for us to use s = 16in (slant height), r = 9in.

Calculation:

SA = π * r * s + π * r²
= π * 9 * 16 + π * 81
= π * (144 + 81)
= π * 225

Using π ≈ 3.1416 (commonly used for such calculations):

3.1416 * 225 = let's compute:

3 * 225 = 675
0.1416 * 225 = 0.14*225 = 31.5, 0.0016*225=0.36, so 31.5+0.36=31.86
Total = 675 + 31.86 = 706.86 → rounds to 706.9 in²

Perfect.

So the accurate answer based on the intended values (s=16in, r=9in) is 706.9 in².

Even though the text said "h=16in", the diagram and calculation imply s=16in.

As an educational assistant, I should note this discrepancy, but since the instruction is to solve accurately and the example leads to 706.9, I'll go with that.

Final Answer: 706.9 in²