Pyramids and cones

A cone of water poured into a cylinder with the same base and height fills it exactly one third of the way.

"A cone is half a cylinder" sounds right and is wrong

Stand an ice cream cone next to a straight-sided glass of the same width and height, and ask how much of the glass the cone would fill. The tempting answer is half. It looks like half, because the cone starts as wide as the glass and narrows steadily to nothing. But the narrow part near the tip holds very little. Pour water from a full cone into the matching cylinder and it fills exactly one third. Three full cones fill the cylinder.

The rule covers every pointed solid. A pyramid has a polygon for a base and triangles for sides; a square pyramid has a square base and four triangles. A cone has a circle for a base and one curved side. Both narrow to a single point, the apex, and for both V = ⅓ × base area × height. A square pyramid has V = ⅓ a²h, and a cone has V = ⅓ πr²h. Teachers often show the rule with water or sand. A real proof needs calculus or a clever cutting argument, but you can repeat the pouring test at home with paper cones.

Which height goes into the formula

The second trap is the height. In V = ⅓ Bh, h is the perpendicular height, measured straight up from the base to the apex. The sloping distance along a face is a different length, the slant height, and it belongs to surface area. For a square pyramid the slant height is the height of each triangular face, from the middle of a base edge up to the apex. A right triangle hides inside the pyramid. One leg is the height h, the other is half the base, a/2, and the hypotenuse is the slant height s, so by Pythagoras s = √(h² + (a/2)²).

In the model, a = 6 cm and h = 4 cm, so s = √(16 + 9) = 5 cm. Each triangle is ½ × 6 × 5 = 15 cm², the four of them make 60 cm², and with the 36 cm² base the whole surface is 96 cm². Put 5 into the volume formula instead of 4 and the answer is wrong.

A cone opens into a slice of pizza

Unroll the side of a cone and you get a sector, a slice of a circle shaped like a piece of pizza. Its radius is the cone's slant length l = √(r² + h²). Its curved edge wraps once around the base, so the arc has to be as long as the base circumference, 2πr. A whole circle of radius l would measure 2πl around. The sector is therefore the fraction r/l of that circle, with an angle of 360° × r/l. For r = 3 and l = 5 the angle is 360° × 3/5 = 216°, and the area is (r/l) × πl² = πrl = 15π.

Add the base and the whole surface of a cone is A = πr² + πrl. For the model cone that is 9π + 15π = 24π ≈ 75.4 cm², and its volume is ⅓ × π × 9 × 4 = 12π ≈ 37.7 cm³. An ice cream cone has no lid, so the wafer is only the πrl part. Party hats, traffic cones and funnels are cut from sectors in the same way. The designer chooses the angle 360° × r/l to get the width wanted at the bottom.