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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.

Contoh soal · volume of a cone with radius 3 height 4

Geometry · Solids

Volume of a cone with radius 3 height 4

Langkah demi langkah

  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.

Tampilkan jawaban
V = 12 \pi \approx 37.699,\quad S = 24 \pi

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Bagaimana: 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.

Pertanyaan yang sering diajukan

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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.

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