Surface Area

Surface Area of a Cuboid #

Add up the net: three pairs of faces.

Surface area is just the area of the net, added up

Volume asks what fits inside. Surface area asks how much paper covers the outside.

Unfold it. Now every face is flat, and you can measure each one.

Three matching pairs: 4 by 3, 4 by 2, and 3 by 2. Six faces, three areas.

12 + 8 + 6 is 26, and every face has a twin: 2 × 26 = 52 in all.

Surface Area of a Cylinder #

Two circles and one unrolled rectangle.

A cylinder unrolls into two circles and one rectangle

A tin has a top, a bottom, and the curved label wrapped round it.

Peel the label off and it is a rectangle. Its width is the circumference, $2\pi r$2πr.

So add two circles to that rectangle and the whole surface is covered.

The Slant Height of a Cone #

Radius and height meet square inside the cone.

The slant of a cone is the hypotenuse of the triangle its radius and height make

A cone is measured two ways: straight up the middle, and along the sloping side.

Stand the height up from the center and run the radius out. They meet at a right angle.

So the slant comes from Pythagoras on the radius and the height — nothing new.

Radius 3 and height 4 give a slant of 5 — a triple you already know.

Cones and Spheres #

The two curved surfaces worth knowing.

A cone and a sphere each have one curved surface with a formula worth knowing

A cone is a base circle of radius r and a curved side of slant length l.

Cut it up the slant and roll it flat: a sector of a circle of radius l.

Its arc is the base rim, $2\pi r$2πr, out of the full $2\pi l$2πl. So it is $\frac{r}{l}$r/l of the circle.

$\frac{r}{l}$r/l of $\pi l^2$πl² is $\pi rl$πrl — the curved part. Add the base circle $\pi r^2$πr² for the whole.

A sphere cannot be flattened at all. Its surface is $4\pi r^2$4πr² — the proof comes later.

Combined Volume and Surface Area #

What fills it, and what covers it.

Volume measures what fills a solid; surface area measures what covers it

A cylinder has radius 3 and height 10. Its volume is $\pi r^{2}h=\pi \times 9\times 10=90\pi$πr²h = π × 9 × 10 = 90π cm³.

Surface area covers it: two circles of $9\pi$9π each, plus the curved surface $60\pi$60π: $78\pi$78π cm².

In decimals, $90\pi$90π cm³ is about 283 cm³ — and each cubic centimeter holds a milliliter.