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

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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

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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

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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

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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

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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.