L4 · Surface Area of Right Cones

Learning goal

“I can find the surface area of a right cone: SA = πr² + πrs, where s is the SLANT height - and I can find s by Pythagoras from the radius and the vertical height when it is not given.”

Right cone — a round solid with a circular base and an apex directly above the centre; its side unrolls into a sector of a circle.

Slant height (s) — the distance from the apex down the curved surface to the edge of the base. It is the radius of the unrolled sector, and it is the height-like length in πrs.

Vertical height (h) — the straight-down distance from apex to base centre, inside the cone. Volume (Lesson L7) uses h; surface area uses s.

Lateral surface area of a cone — the curved part alone: πrs. Think ice-cream cone, party hat, lampshade (open cones have no base circle at all).

Worked example

A right cone has radius 3 cm and slant height 5 cm. Find its total surface area. Step 1: Base: π x 3² = 9π cm2. Step 2: Curved side: πrs = π x 3 x 5 = 15π cm2. Step 3: SA = 9π + 15π = 24π ≈ 75.40 cm2. Where did the 5 come from? Unroll the cone: the side becomes a sector whose radius is the slant height - the length from the apex down the surface to the rim. That is why s, not h, appears in the formula.

Check yourself — explain it

Why does the cone's lateral term use the slant height and not the vertical height?

Paired problem — example still visible

Cone, r = 6 cm, s = 11 cm -> base 36π + side π x 6 x 11 = 66π; SA = 102π ≈ 320.44 cm2

Faded completion — last step is yours

"Cone, r = 4 cm, s = 6 cm: SA = 16π + π x 4 x ___ = 16π + 24π = ___π ≈ ___ cm2" -> 6; 40π ≈ 125.66 cm2

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Grade 10 — Mathematics 10C › C10U02 Surface Area & Volume

L4 · Surface Area of Right Cones