Area & Volume

Let $b$ be a base and $h$ the matching perpendicular height.

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• Rectangle (length $\ell$, width $w$):   $P = 2\ell + 2w$,   $A = \ell w$.
• Triangle:   $P = a+b+c$ (add the three sides),   $A = \tfrac{1}{2}bh$.
• Parallelogram:   $P = 2a + 2b$,   $A = bh$.
• Trapezoid (parallel sides $b_1$ and $b_2$):   $A = \tfrac{1}{2}(b_1+b_2)\,h$.

The height is always measured straight across, at a right angle to the base --- never along a slanted side.

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• Regular polygon (all sides and angles equal): the apothem $a$ is the distance from the center straight out to the middle of a side. With perimeter $P$, @@BLOCK0@@
• Circle with radius $r$ (and diameter $d = 2r$): @@BLOCK1@@
• Sector (a “pizza slice” with central angle $\theta$ degrees): @@BLOCK2@@

A composite figure is made of familiar shapes joined together. To find its area:

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• Cut the figure into shapes you know (rectangles, triangles, half-circles, …).
• Find each piece's area.
• Add the pieces that are present; subtract any region that is cut out.

For perimeter, trace the outside edge only, and do not count any inner cut lines.

Surface area (SA) is the total area of all the outside faces --- the amount of wrapping paper needed. It is always in square units.

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• Prism (base area $B$, base perimeter $P$, height $h$):   $SA = 2B + Ph$.
• Cube (edge $s$):   $SA = 6s^2$ (six equal square faces).
• Cylinder (radius $r$, height $h$):   $SA = 2\pi r^2 + 2\pi r h$.
• Pyramid (base area $B$, base perimeter $P$, slant height $\ell$):   $SA = B + \tfrac{1}{2}P\ell$.
• Cone (radius $r$, slant height $\ell$):   $SA = \pi r^2 + \pi r \ell$.
• Sphere (radius $r$):   $SA = 4\pi r^2$.

The slant height $\ell$ runs up the slanted face of a pyramid or cone --- it is longer than the straight-up height.

A cylinder.

Volume is how much space fills the inside --- always in cubic units.

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• Prism or cylinder:   $V = Bh$, where $B$ is the base area and $h$ the height. itemize
• Rectangular prism: $V = \ell w h$.   Cube: $V = s^3$.   Cylinder: $V = \pi r^2 h$.

Pyramid or cone:   $V = \tfrac{1}{3}Bh$ (exactly one-third of the matching prism or cylinder).

• Cone: $V = \tfrac{1}{3}\pi r^2 h$.

Sphere:   $V = \tfrac{4}{3}\pi r^3$. itemize

A cylinder.

If you multiply every length of a figure by the same factor $k$:

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• Perimeter (a length) is multiplied by $k$.
• Area is multiplied by $k^2$.
• Volume is multiplied by $k^3$.

So doubling the sides ($k=2$) makes area $4\times$ and volume $8\times$ bigger.

Sometimes you know a triangle's three sides but not its height. Heron's formula finds the area directly. First compute the semiperimeter (half the perimeter):

Then the area is

This always works for a valid triangle, and it never needs a perpendicular height.

Half of a rectangle with the same base and height.

When a polygon is given by the coordinates of its corners, two tools find its area without any height at all.

Shoelace formula. List the vertices in order (going around once) as $(x_1,y_1),(x_2,y_2),\dots,(x_n,y_n)$, then loop back to the first. Multiply “down the diagonals” and subtract:

It is called the shoelace because the cross-multiplications criss-cross like laces.

Pick's theorem. If every corner sits on a grid (lattice) point, count $I$, the interior grid points, and $B$, the grid points on the boundary. Then

Two very different counts, one exact area --- and they must agree.

A combined (composite) solid is glued from familiar solids: a cylinder capped by a hemisphere, an ice-cream cone (cone $+$ hemisphere), a house (prism $+$ pyramid), and so on. Treat it like a composite figure, but in $3$D:

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• Volume: add the volumes of the separate pieces. Nothing is lost where they join.
• Surface area: add only the surfaces that are exposed on the outside. The circle (or face) where two pieces meet is inside the solid --- do not count it.

So for a cone stacked on a hemisphere of the same radius, the exposed surface is the cone's slant side $\pi r \ell$ plus the hemisphere's curved half $2\pi r^2$ --- neither flat base circle is counted, because they are hidden at the join.

Cross-section: the flat shape you see when you slice straight through a solid. Slicing a cylinder horizontally gives a circle; slicing it vertically through the middle gives a rectangle. Slicing a cube can give a square, a rectangle, or even a triangle or hexagon, depending on the angle.

Net: a solid unfolded flat, showing every face at once. A cube's net is $6$ squares; a cylinder's net is two circles plus a rectangle whose width equals the circumference $2\pi r$. A net is the fastest way to see why a surface-area formula adds up --- the total area of the net is the surface area.

Scaling: multiply every length by a factor $k$ and lengths grow by $k$, areas by $k^2$, and volumes by $k^3$. A subtle consequence: the surface-area-to-volume ratio scales like $k^2/k^3 = 1/k$, so bigger objects have relatively less surface --- which is why large animals hold heat and small ones lose it fast.