Math · Geometry
Volume and surface area calculator.
Pick a shape and enter its dimensions in any one unit. Results are in that unit cubed (volume) or squared (area).
A sphere with radius 3 has volume (4/3)π(3)³ = 113.097336 and surface area 4π(3)² = 113.097336 (the two happen to match at r = 3). A cone with radius 3 and height 4 has slant height 5, volume 37.699112 and total surface area 75.398224.
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The formulas
Sphere: V = (4/3)πr³, surface area 4πr². Cylinder: V = πr²h, total surface area 2πr(r + h). Cone: V = (1/3)πr²h, total surface area πr(r + l) where the slant height l = √(r² + h²). Pyramid: V = (1/3) × base area × height; this calculator takes a rectangular base with the apex directly above its centre and adds the four triangular faces to the base for the surface area. Trapezoid: area = (a + b)h / 2. Ellipse: area = πab. Sector: area = (1/2)r²θ with θ in radians, and arc length rθ.
The exact perimeter of an ellipse needs an elliptic integral, so the calculator shows Ramanujan's approximation, π[3(a + b) − √((3a + b)(a + 3b))].
Worked example
Sphere, r = 3: V = 36π = 113.097336 and A = 36π = 113.097336. Cone, r = 3, h = 4: l = 5, V = 12π = 37.699112, lateral area 15π = 47.12389, total 24π = 75.398224. Cylinder, r = 2, h = 5: V = 62.831853 and total surface area 87.964594. Square pyramid with a 6 by 6 base and height 4: V = 48 and surface area 96. Trapezoid with sides 5 and 9 and height 4: area 28. Ellipse with semi-axes 5 and 3: area 47.12389, perimeter about 25.526986. Sector with r = 6 and a 60° angle: area 18.849556 and arc length 6.283185.
Limits
Shapes are the right (upright) forms. Use the same unit for every dimension. Answers are rounded to six decimals.
Sources, opened 15 October 2026: Wikipedia, Sphere (V = (4/3)πr³ and surface area 4πr²); Wikipedia, Cone (V = (1/3)πr²h and total surface area πr(r + √(r² + h²))); Wikipedia, Cylinder (V = πr²h and surface area 2πr(r + h)); Wikipedia, Pyramid (V = (1/3)Bh); Wikipedia, Ellipse (area πab and Ramanujan perimeter approximation); Wikipedia, Circular sector (sector area (1/2)r²θ); Wikipedia, Trapezoid (area (1/2)(a + b)h).
Frequently asked questions
How do you find the volume of a sphere?
V = (4/3)πr³. For a radius of 3 that is 36π, or 113.097336 cubic units.
What is the surface area of a cone?
Total surface area = πr(r + l), where l is the slant height √(r² + h²). A cone with r = 3 and h = 4 has l = 5 and a total surface area of 75.398224.
How do you calculate the volume of a cylinder?
V = πr²h. For r = 2 and h = 5 the volume is 62.831853 cubic units.
How is the volume of a pyramid calculated?
One third of the base area times the height. A pyramid with a 6 by 6 base and height 4 has volume (1/3)(36)(4) = 48.
Is the ellipse perimeter exact?
No. There is no simple exact formula, so the calculator reports Ramanujan's approximation, which is very accurate for most ellipses.
Learning aid
Educational calculator with worked steps. Check answers and the required method against your own course or textbook. Nothing you type leaves your browser.