Calculate volume, surface area, and diagonal of 8 common 3D shapes: sphere, cube, cylinder, cone, rectangular prism, triangular prism, pyramid, and torus. Compare shapes side by side.
Solid geometry is the branch of mathematics that studies three-dimensional figures — their volumes, surface areas, and spatial relationships. While plane geometry deals with flat shapes like triangles and circles, solid geometry extends into the third dimension with shapes we encounter daily: boxes (rectangular prisms), cans (cylinders), balls (spheres), ice cream cones, rooftops (pyramids), and doughnuts (tori).
Each 3D shape has characteristic formulas. A sphere of radius r has volume 4πr³/3 and surface area 4πr². A cube of side s has volume s³ and total surface area 6s². A right circular cylinder with radius r and height h has volume πr²h and total surface area 2πr(r + h). A cone with radius r, height h, and slant height l = √(r² + h²) has volume πr²h/3 and lateral surface πrl. Rectangular prisms, triangular prisms, square pyramids, and tori each have their own set of formulas.
This calculator supports eight fundamental 3D shapes through a single dropdown. Enter the relevant dimensions — radius, side length, height, width, depth — and instantly see volume, surface area (total and lateral where applicable), and the space diagonal. A comparison table shows all eight shapes evaluated with the same characteristic dimension, letting you see how volumes and surface areas scale across different geometric solids. Presets and visual bars make exploring 3D geometry intuitive, whether you're a student, engineer, architect, or hobbyist.
Solid geometry problems often require you to remember a different formula set for each shape, keep track of which dimensions are actually needed, and avoid mixing total surface area with lateral surface area. This calculator removes that bookkeeping. You can switch from a sphere to a torus or prism without rebuilding the problem from scratch, and the output cards make it clear which measurements belong to the selected solid.
It is especially useful when you need to compare shapes rather than solve a single isolated exercise. Students can verify homework, teachers can demonstrate how volume and area scale differently, and designers can quickly test which solid gives more capacity or less outer surface for similar dimensions.
Sphere: V = 4πr³/3, SA = 4πr² Cube: V = s³, SA = 6s², d = s√3 Cylinder: V = πr²h, SA = 2πr(r+h) Cone: V = πr²h/3, SA = πr(r+l), l = √(r²+h²) Rectangular prism: V = lwh, SA = 2(lw+lh+wh), d = √(l²+w²+h²) Triangular prism: V = (base area)×h, SA = (base perimeter)×h + 2×(base area) Pyramid (square): V = s²h/3, SA = s² + 2sl Torus: V = 2π²Rr², SA = 4π²Rr
Result: Volume ≈ 785.40, Total SA ≈ 471.24, Lateral SA ≈ 314.16
For a cylinder with radius 5 and height 10: V = π(5²)(10) = 250π ≈ 785.40. Lateral SA = 2π(5)(10) = 100π ≈ 314.16. Total SA = 2π(5)(5+10) = 150π ≈ 471.24.
A useful habit in solid geometry is to separate questions about capacity from questions about material. Volume tells you how much space a shape encloses, while surface area tells you how much outer covering it needs. Two solids can have similar overall dimensions but very different efficiency. A sphere, for example, encloses far more volume per unit of surface area than a box with the same characteristic length, while a torus spreads material over a more complex outer boundary. Looking at both values together helps you understand why some shapes are chosen for storage, packaging, or structural design.
Each solid in this calculator has one or two details that students often miss. Cones and pyramids use slant height in surface-area formulas, but vertical height in volume formulas. Rectangular prisms and cubes have space diagonals that cut through the interior, not just across a face. Triangular prisms require the base triangle area before multiplying by prism length. Tori need both a major radius and a minor radius, and confusing those values changes the geometry completely. Seeing all outputs at once helps you catch those mismatches before they turn into larger errors.
This kind of tool is most helpful when you are exploring alternatives instead of plugging numbers into a single memorized formula. If you are studying for an exam, you can move through several solids and see how the equations change. If you are estimating materials, you can compare which shape gives the best balance between volume and exposed area. If you are teaching, the shared comparison table makes scale effects visible immediately: doubling a dimension multiplies surface area by four but volume by eight.
Eight shapes: sphere, cube, cylinder, cone, rectangular prism, triangular prism, square pyramid, and torus. Use this as a practical reminder before finalizing the result.
V = πr²h/3, where r is the base radius and h is the height. It's exactly one-third of the cylinder with the same base and height.
It's the longest line segment inside the box: d = √(l² + w² + h²). For a cube of side s, this simplifies to s√3.
A torus is a doughnut shape defined by major radius R (center to tube center) and minor radius r (tube radius). Volume = 2π²Rr².
The sphere — it encloses the maximum volume for any given surface area, a fact proven by the isoperimetric inequality in 3D. Keep this note short and outcome-focused for reuse.
Lateral SA is only the "side" surface (excluding bases/ends). Total SA includes the bases. For a cylinder: lateral = 2πrh, total adds the two circular caps 2πr².