A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (Use $\pi = \frac{22}{7}$ )

Surface Areas and Volumes

Volume of a Combination of Solids

Official Answer

Total volume ≈ $680.17\text{ cm}^3$ (exact value $\frac{4081}{6}\text{ cm}^3$ ). With radius $r = 3.5\text{ cm}$ , cylinder height $h = 20-7 = 13\text{ cm}$ ; $\text{Volume} = \pi r^2 h + \frac{4}{3}\pi r^3 = 500.5 + 179.67$ ≈ $680.17\text{ cm}^3$ .

cylinder with hemispherical endscombination of solidsvolume of cylindervolume of hemispherecapsule shapemensuration

Marking Scheme

11 mark: correctly finding radius $r = 3.5\text{ cm}$ and cylinder height $h = 20 - 2(3.5) = 13\text{ cm}$ .
21 mark: correct formula setup — $\text{Total Volume} = \pi r^2 h + 2\times\frac{2}{3}\pi r^3$ (cylinder + two hemispherical ends).
31 mark: correct substitution and final numeric answer ≈ $680.17\text{ cm}^3$ (or equivalent exact fraction $\frac{4081}{6}\text{ cm}^3$ ), with proper units.

Hint

$\text{Radius} = \frac{\text{diameter}}{2} = 3.5\text{ cm}$ ; height of cylinder = total height − 2×radius; then $\text{Volume} = \pi r^2 h \text{ (cylinder)} + 2\times\frac{2}{3}\pi r^3 \text{ (two hemispheres)}$ .

Quick Oral Answer

Find radius 3.5 cm from diameter 7 cm, subtract 2×radius from total height 20 cm to get cylinder height 13 cm, then add the cylinder's volume $\pi r^2 h$ to the volume of two hemispheres $\frac{2}{3}\pi r^3$ each, giving about 680.17 cm³.

Analysis & Explanation

A combination-solids mensuration problem: a cylinder capped by two hemispheres, forming a capsule shape.

Concept

Each hemisphere adds height equal to its radius (not diameter), so $\text{Total height} = \text{cylinder height} + 2r$ .
$\text{Total Volume} = \text{Volume of cylinder} + 2\times\text{Volume of hemisphere} = \pi r^2 h + \frac{4}{3}\pi r^3$ .

Key points

Radius $r = \frac{\text{diameter}}{2} = 3.5\text{ cm}$ ; cylinder height $h = 20 - 2(3.5) = 13\text{ cm}$ .
Use $\pi = \frac{22}{7}$ throughout, keeping fractions exact until the final step to avoid rounding drift.

Common mistakes

Subtracting 2×diameter instead of 2×radius from the total height.
Rounding intermediate hemisphere volumes too early, causing accumulated error.

Real-world

This capsule shape models medicine capsules, storage tanks and test tubes with domed ends.

Common Mistakes

1Subtracting 2× the diameter instead of 2× the radius from the total height to find the cylinder's height, leading to a negative or wrong cylinder height.
2Using the full sphere volume formula $\frac{4}{3}\pi r^3$ instead of the hemisphere formula $\frac{2}{3}\pi r^3$ for each rounded end.
3Arithmetic slips when multiplying fractions with $\pi=\frac{22}{7}$ across multiple terms, especially forgetting to multiply by 2 for the two hemispherical ends.

Interesting Facts

This 'cylinder with hemispherical ends' shape is called a capsule or 'spherocylinder' in geometry, and is exactly the shape of medicine capsules, biological cells like certain bacteria (e.g., E. coli), and pressure vessels in engineering.

Designing storage tanks and pressure vessels with hemispherical (domed) ends rather than flat ends is a real engineering practice because domed ends distribute internal pressure more evenly, reducing stress concentration.

The volume formula for this composite solid — $\pi r^2\left(h + \frac{4r}{3}\right)$ — reduces to the volume of a pure sphere $\frac{4}{3}\pi r^3$ if the cylindrical height h is set to zero, showing the sphere is a limiting case of this shape.

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Frequently Asked Questions

Why do we subtract 2×radius (not 2×diameter) from the total height?

Because each hemispherical end adds a height equal to its radius (a hemisphere's 'height' from its flat circular base to its curved top equals the radius), not its diameter. So $\text{Total height} = \text{cylinder height} + \text{radius (top)} + \text{radius (bottom)} = \text{cylinder height} + 2r$ .

What is the general formula for the volume of this composite solid?

$\text{Total Volume} = \pi r^2 h + \frac{4}{3}\pi r^3 = \pi r^2\left(h + \frac{4r}{3}\right)$ , where r is the common radius of the cylinder and hemispheres, and h is the height of only the cylindrical part.