Cones and Spheres

The surface of a cone

The surface of a cone has two parts: the flat circular base, of radius r, and the curved surface that rises from the rim of the base to the apex. The base is a circle, so its area is $\pi r^2$πr². The curved surface needs the slant height, l, the distance from the apex down the side to the rim.

The slant height is the hypotenuse of the right-angled triangle made by the radius and the perpendicular height, h, so $l^2=r^2+h^2$l² = r² + h². Take a cone with a radius of 4 cm and a slant height of 10 cm. Its height is $\sqrt{10^2-4^2}=\sqrt{84}$√(10² − 4²) = √84, about 9.17 cm, but its surface needs only the radius and the slant height.

A cone with base radius r and slant height l.

Roll the curved surface flat

Cut the curved surface along a straight line from the apex down to the rim, and roll it out flat. It lies flat without stretching or tearing, and it makes a sector: the part of a circle between two radii, like a slice of a round cake.

Every point of the rim was a distance l from the apex, and the apex becomes the center of the sector. So the sector is cut from a circle of radius l, and its curved edge, the arc, is what was the rim of the base.

The curved surface laid flat: a sector of a circle of radius l, whose arc is as long as the rim of the base, $2\pi r$2πr. For the cone with r = 4 and l = 10 it is $\frac{2}{5}$2/5 of the circle, 144°.

What fraction of the circle

The arc was the rim of the base, so its length is the circumference of the base, $2\pi r$2πr. The whole circle of radius l has circumference $2\pi l$2πl. So the arc is $\frac{2\pi r}{2\pi l}=\frac{r}{l}$2πr/2πl = r/l of the whole circumference.

The sector is the same fraction of the whole circle. Cut the circle into equal sectors, and each has an equal share of both the circumference and the area; take a sector with twice the arc, and it is two of those pieces, with twice the area. So the sector is $\frac{r}{l}$r/l of the circle.

For the cone with a radius of 4 cm and a slant height of 10 cm, the sector is $\frac{4}{10}=\frac{2}{5}$4/10 = 2/5 of a circle of radius 10 cm, and its angle is $\frac{2}{5}\times 360^{\circ }=144^{\circ }$2/5 × 360° = 144°.

$\pi rl$πrl, and the base

The whole circle of radius l has area $\pi l^2$πl². The sector is $\frac{r}{l}$r/l of it: $\frac{r}{l}\times \pi l^2=\pi rl$r/l × πl² = πrl. So the curved surface area of a cone is $\pi rl$πrl, and the total surface area of a cone, with its base, is $\pi rl+\pi r^2$πrl + πr².

For the cone with a radius of 4 cm and a slant height of 10 cm, the curved surface is $\pi \times 4\times 10=40\pi$π × 4 × 10 = 40π cm², which is $\frac{2}{5}$2/5 of the circle’s $\pi \times 10^2=100\pi$π × 10² = 100π cm². The base adds $\pi \times 4^2=16\pi$π × 4² = 16π cm², so the total is $56\pi$56π cm², about 175.9 cm².

A cone with a radius of 5 cm and a slant height of 9 cm has a curved surface of $\pi \times 5\times 9=45\pi$π × 5 × 9 = 45π cm², about 141.4 cm². With its base, $25\pi$25π cm², the total is $70\pi$70π cm², about 219.9 cm².

The surface of a sphere

A sphere cannot be laid flat without stretching or tearing, however it is cut; the peel of an orange splits when it is pressed flat. So its surface area needs a formula of its own: a sphere of radius r has surface area $4\pi r^2$4πr², four times the area of a circle with the same radius. A sphere with a radius of 5 cm has a surface area of $4\times \pi \times 5^2=100\pi$4 × π × 5² = 100π cm², about 314.2 cm².

Archimedes proved this. His proof is long, but the volume of the sphere gives a good reason to believe it. Cover the sphere’s surface with many tiny patches, and join the edges of each patch to the center. That cuts the sphere into many thin pieces, each very nearly a pyramid whose base is a patch and whose height is the radius, r. A pyramid holds ⅓ × base × height, so all the pieces together hold ⅓ × r × (the total area of the patches), which is ⅓ × r × the surface area S.

That total is the volume of the sphere, $\frac{4}{3}\pi r^3$4/3 πr³. So $\frac{1}{3}\times r\times S=\frac{4}{3}\pi r^3$⅓ × r × S = 4/3 πr³, and multiplying both sides by 3 and dividing by r gives $S=4\pi r^2$S = 4πr².

Archimedes found the same answer another way: the surface of a sphere has exactly the same area as the curved surface of the cylinder that fits snugly around it. That cylinder has radius r and height 2r, so its curved surface is $2\pi r\times 2r=4\pi r^2$2πr × 2r = 4πr².

The sphere and the cylinder that fits snugly around it. The sphere’s surface has the same area as the cylinder’s curved side, $2\pi r\times 2r=4\pi r^2$2πr × 2r = 4πr².

A hemisphere

A hemisphere is half a sphere. Its curved surface is half of $4\pi r^2$4πr², which is $2\pi r^2$2πr², and a solid hemisphere also has a flat circular face of area $\pi r^2$πr², so its total surface area is $3\pi r^2$3πr².

Three slips

Writing $2\pi rl$2πrl, as if the cone’s curved surface were a cylinder’s. A cone’s curved surface unrolls into a sector, not a rectangle, and the sector is $\frac{r}{l}$r/l of the circle $\pi l^2$πl², which is $\pi rl$πrl, with no 2.

Using the perpendicular height in place of the slant height. For the cone with r = 3 cm and h = 4 cm, $\pi \times 3\times 4=12\pi$π × 3 × 4 = 12π cm² is too small; the curved surface runs along the slope, so it is $\pi \times 3\times 5=15\pi$π × 3 × 5 = 15π cm².

Mixing up the sphere’s two formulas. An area is measured in square units and has $r^2$r², so it is $4\pi r^2$4πr²; the volume is in cubic units and has $r^3$r³, $\frac{4}{3}\pi r^3$4/3 πr³.