A right circular cone has a height of 12 cm and a radius of 5 cm. Find its slant height and curved surface area.
13 cm and 204.2 cm²
The three quantities of a right circular cone — slant height \(l\), vertical height \(h\) and base radius \(r\) — form a right triangle, so they are linked by \(l^2 = h^2 + r^2\).
Substituting the given values \(h = 12\text{ cm}\) and \(r = 5\text{ cm}\): \(l = \sqrt{12^2 + 5^2}\).
Evaluating inside the root: \(l = \sqrt{144 + 25} = \sqrt{169} = 13\text{ cm}\). This is the well-known 5-12-13 Pythagorean triple.
The curved (lateral) surface area of a cone is given by \(\text{CSA} = \pi r l\), the area of the sloping surface only (it excludes the flat circular base).
Putting the numbers in: \(\text{CSA} = \pi \times 5 \times 13 = 65\pi\text{ cm}^2\).
Taking \(\pi = 3.14\): \(\text{CSA} = 65 \times 3.14 = 204.1 \approx 204.2\text{ cm}^2\).
Key concept: for a cone, \(l = \sqrt{h^2 + r^2}\) and \(\text{CSA} = \pi r l\). Hence the slant height and curved surface area are 13 cm and 204.2 cm².
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