The Capsule Calculator treats a capsule as a cylinder with hemispherical caps on both ends: a pill shape. Enter the shared radius and the height of the cylindrical middle section (not including the caps). The page redraws a labelled figure with radius and cylinder height marked.
Volume is cylinder volume plus one full sphere volume (two hemispheres). Surface area is the cylinder wall plus the full sphere surface, with no flat ends. Related pages cover the Cylinder Calculator, the Hemisphere Calculator, and the 2D cousin on the Stadium Calculator.
Calculate the volume of a capsule
Volume of a capsule is the sum of a cylinder of radius r and height h plus a sphere of radius r. The Capsule Calculator uses that decomposition. Radius and cylindrical height must be positive. Total tip-to-tip length is h + 2r when the caps are full hemispheres.
V = pi * r^2 * h + (4/3) * pi * r^3Equivalently, V = pi * r^2 * (h + (4/3) * r). The sphere term does not use h; only the cylindrical middle grows with h. Doubling h adds another pi * r^2 * h to volume without changing the caps.
For r = 3 and h = 10:
V = pi * 9 * 10 + (4/3) * pi * 27 = 90 * pi + 36 * pi = 126 * pi approx 395.84Exact form 126 * pi matches the outline fixture. If a drawing labels overall length L instead of cylinder height, convert with h = L - 2r before using the formula, provided L is greater than 2r.
Calculate the surface area
Surface area of a closed capsule is the lateral cylinder wall plus the full sphere surface. There are no flat circular lids; the hemispheres replace them. The Capsule Calculator reports that closed skin.
SA = 2 * pi * r * h + 4 * pi * r^2Equivalently, SA = 2 * pi * r * (h + 2r). The 4 * pi * r^2 term is exactly the sphere surface at radius r. Open capsules are rare in worksheet problems; this page assumes the closed pill.
For r = 3 and h = 10:
SA = 2 * pi * 3 * 10 + 4 * pi * 9 = 60 * pi + 36 * pi = 96 * pi approx 301.59A bare cylinder with the same r and h has volume 90 * pi and total surface area 78 * pi including two flat ends. The capsule volume is larger by the sphere term 36 * pi, and the surface formula differs because rounded caps replace flat lids.
Solve a capsule with radius 3 and height 10
Use r = 3 and cylindrical section h = 10 as the reference capsule. Overall length is 10 + 6 = 16. Volume and surface area factor with pi. 1. Overall length
L = h + 2*r = 10 + 6 = 162. Volume
V = 90 * pi + 36 * pi = 126 * pi approx 395.843. Surface area
SA = 60 * pi + 36 * pi = 96 * pi approx 301.59Cylinder contribution and sphere contribution are visible as separate addends, which helps check arithmetic against the Cylinder Calculator and Sphere Calculator pages without copying those pages' full treatments.
If a drawing labels overall length L = 16 with radius 3, convert to h = 16 - 2*3 = 10 before using the volume formula. Entering L as if it were h would treat the caps as empty space and under-count volume by the sphere term. Medicine and tank problems often print overall length on the label; worksheet language varies, so read whether height means the middle cylinder or tip-to-tip.
Surface coatings on capsules follow SA = 96 * pi for the reference case. That area is not the cylinder total 78 * pi, because rounded caps replace flat lids and add the sphere skin instead of two disks.
Recover cylinder height from overall length
Medicine and tank drawings often print tip-to-tip length L. Convert with h = L - 2r when L > 2r, then enter h as the cylindrical middle height on the Capsule Calculator. For L = 16 and r = 3, h = 10, which recovers volume 126 * pi and surface area 96 * pi.
If L is less than or equal to 2r, the inputs cannot describe full hemispherical caps on a positive middle section.
Volume splits cleanly for checking: cylinder piece pi * r^2 * h plus sphere piece (4/3) * pi * r^3. For the reference inputs those pieces are 90 * pi and 36 * pi. Changing only h scales the cylinder piece; the sphere piece stays fixed for fixed r.
Unit note: radius and middle height share one length unit. Overall length uses the same unit. Volume is in cube units; closed surface area is in square units. Density times volume estimates mass for a solid capsule blank, but this page returns geometry only.
Understand the capsule and stadium relationship
A stadium (discorectangle) is the 2D silhouette of a capsule: a rectangle with semicircles on opposite ends. The Stadium Calculator owns planar area and perimeter. The Capsule Calculator owns 3D volume and surface area. They share radius and straight-segment length as inputs but not formulas.
Cross-section through the axis of a capsule is a stadium with the same r and straight length h. Area of that stadium is not the capsule volume; volume integrates the circular cross sections along the axis. Keep the pages linked as a family and keep worked examples separate: stadium uses a = 10 and r = 3 for 2D area; capsule uses h = 10 and r = 3 for 3D volume 126 * pi.
Running-track diagrams are stadiums. Medicine-pill solids are capsules. Mixing the words without checking dimension is a common search-intent collision that these two URLs are meant to split.
Frequently asked questions
What is capsule volume?
Capsule volume is pi * r^2 * h + (4/3) * pi * r^3. For r = 3 and h = 10, V = 126 * pi approx 395.84.
What is capsule surface area?
Surface area is 2 * pi * r * h + 4 * pi * r^2. For r = 3 and h = 10, SA = 96 * pi approx 301.59.
Does height include the caps?
No. Enter the cylindrical middle height only. Overall length is h + 2r.
How does a capsule relate to a stadium?
A stadium is the 2D capsule silhouette. Use the Stadium Calculator for area and perimeter; use this page for 3D volume and surface area.
How does capsule volume relate to a cylinder and sphere?
Volume is exactly cylinder volume plus sphere volume at the same radius, with cylinder height equal to the middle section.
Can the Capsule Calculator start from overall length?
Yes. Convert L to h = L - 2r when L > 2r, then apply the volume and surface formulas.
What units does the Capsule Calculator use?
Radius and height share one length unit. Volume uses cube units; surface area uses square units.
Are the caps always hemispheres?
This page assumes full hemispherical caps of the same radius as the cylinder. Partial caps need spherical-cap formulas instead.
What is the reference capsule volume?
For r = 3 and cylindrical height 10, volume is 126 * pi approx 395.84.
How do I convert overall length to cylinder height?
Use h = L - 2r when overall length L is greater than 2r. For L = 16 and r = 3, h = 10.
Summary
Capsule Calculator returns volume and surface area for a cylinder with hemispherical caps from radius and middle height, with substituted arithmetic and a labelled figure. Worked numbers match r = 3 and h = 10: V = 126 * pi and SA = 96 * pi.
Use the Stadium Calculator for the 2D outline and the Cylinder or Hemisphere calculators for the separate pieces.