Polygons and Solids

Special Quadrilaterals #

Named by what stays equal.

Each special quadrilateral is named by which sides and angles it keeps equal

A rectangle keeps four right angles and opposite sides equal.

A parallelogram keeps opposite sides parallel but lets the corners lean.

A trapezium keeps just one pair of sides parallel.

Rhombus and Kite #

Equal sides, two different ways.

A rhombus keeps all four sides equal and a kite keeps two pairs of equal neighbors

A rhombus is a leaning square: four equal sides, but its corners need not be 90°.

A rhombus’s two diagonals always cross at right angles, and each cuts the other in half.

A kite is two pairs of equal neighboring sides. It folds flat along one diagonal.

Its diagonals cross at right angles too — but only the fold line cuts the other in half.

Area of Composite Shapes #

Simple shapes added or taken away.

A complicated outline is just simple shapes added or taken away

Start with the whole rectangle, 8 by 5. That is 40 squares.

This corner is the piece being taken out: 3 across and 2 down, so 6 squares.

Take it away and the step is left behind. 40 − 6 = 34 squares in the L-shape.

Prisms and Cylinders #

Cross section times the length.

A prism holds the same cross section all the way along its length

Cut straight across this prism anywhere, and the face you expose is the same rectangle.

That face is 12 and the prism runs 5 along, so the volume is 12 × 5 = 60.

A cylinder works the same way — its cross section is a circle, $\pi r^2$πr², so the volume is $\pi r^{2}h$πr²h.

Pyramids and Cones #

Exactly a third of the box around it.

A pyramid or cone holds exactly $\frac{1}{3}$1/3 of the prism around it

A pyramid sits inside a box on the same base, rising to the same height.

A cone fits its cylinder the same way: three cone-fulls of water fill the cylinder.

Either way the solid holds $\frac{1}{3}$1/3 of its container: $\frac{1}{3}\times$1/3 × base area × height.

Spheres #

Only the radius decides anything.

The volume of a sphere is $\frac{4}{3}\pi r^3$4/3 πr³

Every point of the surface sits the same distance from the center: the radius r.

Fit it snugly in a cylinder, radius r and height 2r. It fills $\frac{2}{3}$2/3 of it.

The cylinder is $2\pi r^3$2πr³, and $\frac{2}{3}$2/3 of that is $\frac{4}{3}\pi r^3$4/3 πr³. The proof of the $\frac{2}{3}$2/3 comes later.

Plans and Elevations #

Three flat views pin down one solid.

The plan and the two elevations are the flat views of a solid from above, in front and beside

One solid, three flat views: from the top, from the front, and from the side.

The plan is the view straight down — each square is a column seen from above.

The front elevation faces the solid head on: height shows, depth disappears.

Together the three views usually pin the solid down. One view alone never can.

Drawing Solids on a Dot Grid #

Every edge one hop, along one of three directions.

On isometric dots, a solid’s edges each run one hop along one of three directions

A cube on isometric dots: every edge is one hop, along one of three directions.

Stretch one direction two hops and the sketch is a cuboid, two cubes long.

Scaling Area and Volume #

Double the length, eight times the volume.

Doubling every length multiplies area by four and volume by eight

Double every length. The front face now holds 2 × 2 = 4 of the small squares.

Behind that face sits a second layer, so 2 × 4 = 8 small cubes fill it.

Triple it instead: 3 × 3 = 9 squares on the face, 3 × 9 = 27 cubes inside.

Euler’s Formula for Polyhedra #

Corners minus edges plus faces is always two.

For every solid with flat faces, corners − edges + faces = 2

A cube has 8 corners: seven you can see, and one hidden behind the others.

The same cube has 12 edges — nine drawn solid, three dashed where they run behind.

And 6 faces: three turned toward you, three away. So 8 − 12 + 6 = 2.

Stretch it into a cuboid: the same 8 corners, only further apart.

A square pyramid is a different solid, and 5 − 8 + 5 = 2 all the same.

Why There Are Exactly Five Platonic Solids #

The angles at a corner run out of room.

Only five solids can be built from one repeated regular face

A corner of a solid needs three faces at least: two alone only fold flat.

Six triangles round a point fill 360° and lie flat, so a corner needs less.

Under 360° a corner takes 5 triangles at most, 3 squares, 3 pentagons, 0 hexagons.

Count the ways: 3, 4 or 5 triangles, 3 squares, 3 pentagons — the five Platonic solids.

Their names, in that order: tetrahedron, octahedron and icosahedron from triangles, the cube from squares, the dodecahedron from pentagons.

Archimedes and the Volume of a Sphere #

A hemisphere is a cylinder minus a cone.

A hemisphere fills exactly what is left of a cylinder when a cone is drilled out

The Spheres lesson gave you $\frac{4}{3}\pi r^3$4/3 πr³ without proof. Archimedes supplies the proof.

A hemisphere, and beside it a cylinder of the same r with a cone drilled out.

Slice both at any height h. Slide the cut, and watch the two cross-sections.

The slice edge is r from the center, h up: by Pythagoras its radius squared is $r^2-h^2$r² − h².

The ring is $\pi r^2-\pi h^2$πr² − πh², the same $\pi \left(r^2-h^2\right)$π(r² − h²). Equal slices, so equal volumes.

Cylinder $\pi r^3$πr³ minus cone $\frac{\pi r^3}{3}$πr³/3 leaves $\frac{2}{3}\pi r^3$2/3 πr³. Double that for the whole sphere.