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Solid geometry: prisms, pyramids, spheres

Volumes, surface areas and cross-sections in three dimensions.

Prisms and cylinders are base × height; pyramids and cones a third of that; spheres 4/3 πr³. Picture it: a cone fits three times into the cylinder around it — rotate the 3D view to see the base and slant. Think it: Cavalieri's principle — equal cross-sections at every height means equal volume — is the idea calculus later makes exact.

Isibonelo esisebenza: volume of a cone with radius 3 height 4

Geometry · Solids

Volume of a cone with radius 3 height 4

Isigaba

  1. r = 3,\ h = 4,\ \ell = \sqrt{r^2+h^2} = 5

    Cone\; ℓ is the slant height.

  2. V = \tfrac{1}{3}\pi r^2 h = 12 \pi \approx 37.699

    Volume — a third of the cylinder around it.

  3. S = \pi r^2 + \pi r \ell = 24 \pi \approx 75.398

    Surface area = base + lateral.

Bonisa impendulo
V = 12 \pi \approx 37.699,\quad S = 24 \pi

Symbols used here

\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
\approx
approximately equal
Equal to the precision shown, not exactly.
\theta
theta
The usual name for an angle.
^\circ
degrees
1/360 of a full turn. 180° = π radians.
\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
\angle ABC,\ \triangle ABC
angle, triangle
The angle at B between BA and BC; the triangle with those vertices.
\parallel,\ \perp,\ \cong,\ \sim
parallel, perpendicular, congruent, similar
Never meet; meet at 90°; identical shape and size; same shape.

How to: Solid geometry: prisms, pyramids, spheres

  1. Cone\; ℓ is the slant height.
  2. Volume — a third of the cylinder around it.
  3. Surface area = base + lateral.

Questions people ask

Why does every triangle have angles adding to 180°?

Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.

When do I use the law of sines versus the law of cosines?

Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.

What is the difference between area and perimeter?

Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.

Zama wena

Okuningi Geometry

TrianglesPythagorean theoremCirclesPolygonsSolidsCoordinate geometryEuclid's axioms and the structure of proofCongruent and similar trianglesCircle theoremsTransformations: translation, rotation, reflection, dilation