Surface area of a cone practice problems with diagrams and measurements.

To find the surface area of a cone, we use the formula:
Surface Area = $\pi r^2 + \pi r s$

Where:
* $r$ is the radius of the base.
* $s$ is the slant height (the diagonal side).
* $\pi$ is approximately $3.14$.

Sometimes you are given the vertical height ($h$) instead of the slant height. In that case, you must use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the slant height first: $s = \sqrt{r^2 + h^2}$.

Here are the step-by-step solutions for each problem:

1) Radius ($r$) = 16 m, Slant Height ($s$) = 19 m
* Base Area: $3.14 \times 16^2 = 3.14 \times 256 = 803.84$
* Lateral Area: $3.14 \times 16 \times 19 = 954.56$
* Total: $803.84 + 954.56 = 1758.40$

2) Diameter = 6 ft, so Radius ($r$) = 3 ft. Slant Height ($s$) = 10 ft.
* Base Area: $3.14 \times 3^2 = 3.14 \times 9 = 28.26$
* Lateral Area: $3.14 \times 3 \times 10 = 94.20$
* Total: $28.26 + 94.20 = 122.46$

3) Radius ($r$) = 4 yd, Slant Height ($s$) = 6 yd
* Base Area: $3.14 \times 4^2 = 3.14 \times 16 = 50.24$
* Lateral Area: $3.14 \times 4 \times 6 = 75.36$
* Total: $50.24 + 75.36 = 125.60$

4) Diameter = 4 yd, so Radius ($r$) = 2 yd. Slant Height ($s$) = 6 yd.
* Base Area: $3.14 \times 2^2 = 3.14 \times 4 = 12.56$
* Lateral Area: $3.14 \times 2 \times 6 = 37.68$
* Total: $12.56 + 37.68 = 50.24$

5) Radius ($r$) = 13 m, Height ($h$) = 37 m. First, find slant height ($s$).
* $s = \sqrt{13^2 + 37^2} = \sqrt{169 + 1369} = \sqrt{1538} \approx 39.22$ m
* Base Area: $3.14 \times 13^2 = 3.14 \times 169 = 530.66$
* Lateral Area: $3.14 \times 13 \times 39.22 \approx 1604.89$
* Total: $530.66 + 1604.89 = 2135.55$

6) Diameter = 15 ft, so Radius ($r$) = 7.5 ft. Slant Height ($s$) = 20 ft.
* Base Area: $3.14 \times 7.5^2 = 3.14 \times 56.25 = 176.625$
* Lateral Area: $3.14 \times 7.5 \times 20 = 471.00$
* Total: $176.625 + 471.00 = 647.625 \rightarrow 647.63$

7) Radius ($r$) = 6 ft, Slant Height ($s$) = 9 ft
* Base Area: $3.14 \times 6^2 = 3.14 \times 36 = 113.04$
* Lateral Area: $3.14 \times 6 \times 9 = 169.56$
* Total: $113.04 + 169.56 = 282.60$

8) Diameter = 8 yd, so Radius ($r$) = 4 yd. Slant Height ($s$) = 12 yd.
* Base Area: $3.14 \times 4^2 = 3.14 \times 16 = 50.24$
* Lateral Area: $3.14 \times 4 \times 12 = 150.72$
* Total: $50.24 + 150.72 = 200.96$

9) Diameter = 11 m, so Radius ($r$) = 5.5 m. Height ($h$) = 14 m. First, find slant height ($s$).
* $s = \sqrt{5.5^2 + 14^2} = \sqrt{30.25 + 196} = \sqrt{226.25} \approx 15.04$ m
* Base Area: $3.14 \times 5.5^2 = 3.14 \times 30.25 = 94.985$
* Lateral Area: $3.14 \times 5.5 \times 15.04 \approx 260.39$
* Total: $94.985 + 260.39 = 355.375 \rightarrow 355.38$

Final Answer:
1) 1758.40 sq m
2) 122.46 sq ft
3) 125.60 sq yd
4) 50.24 sq yd
5) 2135.55 sq m
6) 647.63 sq ft
7) 282.60 sq ft
8) 200.96 sq yd
9) 355.38 sq m