maths.free › Geometry › Solid geometry: prisms, pyramids, spheres

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.

ნამდვილი ასლი · volume of a cone with radius 3 height 4

Geometry · Solids

Volume of a cone with radius 3 height 4

Step by step

  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.

Reveal the answer
V = 12 \pi \approx 37.699,\quad S = 24 \pi

Now you Pick a problem, or type or draw your own. Every step, a picture, the answer hidden until you ask.

Keep your own working

უფასო ანგარიშის გამოყენებისას, ყველა გაკვეთილზე დამატებით ჩანაწერებს ნახავთ, დასრულებული და გადაწყვეტილი ამოცანების ჩანაწერებს ერთ ადგილას და მასწავლებელს, რომელსაც ამ გვერდის შესახებ შეკითხვა შეგიძლიათ გაუწიოთ. მათემატიკა ყველასთვის ხელმისაწვდომია, ანგარიშის შექმნის შემდეგაც კი.

რეგისტრაცია შესვლა

აქ გამოყენებული სიმბოლოები

Tap any symbol for the full definition, a picture, and what every letter in it means.

როგორ: 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.

მეტი Geometry

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