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 $\pi = 22/7$ )

Q: Is the answer 680.17 cm³ exact?

The exact value is $4081/6\text{ cm}^3$, which equals $680.1666...\text{ cm}^3$, usually written as $680.17\text{ cm}^3$ or $680\frac{1}{6}\text{ cm}^3$. Any of these forms is acceptable in the exam as long as the working is shown.

Surface Areas and Volumes

Volume of a combination of solids

Official Answer

$\text{Total volume} = 4081/6\text{ cm}^3 \approx 680.17\text{ cm}^3$ . The cylinder has radius $r = 3.5\text{ cm}$ and height $h = 20 - 7 = 13\text{ cm}$ , giving volume $\pi r^2 h = 500.5\text{ cm}^3$ . The two hemispherical ends combine into one full sphere of volume $\frac{4}{3}\pi r^3 = 539/3 \approx 179.67\text{ cm}^3$ . Adding these gives the total volume $500.5 + 179.67 = 680.17\text{ cm}^3$ (i.e., 4081/6 cm³).

combination of solidscylinder with hemispherical endsvolumeradius 3.5 cmcylinder height 13 cm(4/3)πr³πr²h680.17 cm³

Marking Scheme

11 mark: Correctly identifying $r = 3.5\text{ cm}$ and finding the cylinder height $h = 20 - 7 = 13\text{ cm}$ .
21 mark: Volume of cylinder $\pi r^2 h = 500.5\text{ cm}^3$ (or correct expression) and volume of the two hemispheres/sphere $\frac{4}{3}\pi r^3 = 539/3 \approx 179.67\text{ cm}^3$ .
31 mark: Adding to get the total volume = $4081/6 \approx 680.17\text{ cm}^3$ . Accept 680.17 cm³, $680\frac{1}{6}\text{ cm}^3$ , or the exact fraction $4081/6\text{ cm}^3$ .

Hint

$\text{Radius} = 3.5\text{ cm}$ . Cylinder height = $20 - 2\times 3.5 = 13\text{ cm}$ . Total volume = $\pi r^2 h + \frac{4}{3}\pi r^3$ (two hemispheres make one sphere).

Quick Oral Answer

Radius is 3.5 cm, so the cylinder height is 20 minus 7, that is 13 cm. Volume equals $\pi r^2 h$ plus $\frac{4}{3}\pi r^3$ for the two hemispheres, which comes to 500.5 plus 179.67, about 680.17 cubic centimetres.

Analysis & Explanation

A standard 'combination of solids' mensuration problem: a cylinder capped by two hemispheres.

Concept & key steps

The total height includes both hemispherical caps, so the cylinder's own height = total height − 2 × radius = $20 - 7 = 13\text{ cm}$ .
Two hemispheres of equal radius combine into one full sphere, so their combined volume is $\frac{4}{3}\pi r^3$ rather than doubling $\frac{2}{3}\pi r^3$ separately.
Keeping $r = 7/2$ as a fraction lets the 7 cancel cleanly with $\pi = 22/7$ .

Common mistakes

Taking the cylinder's height as the full 20 cm, forgetting to subtract the two hemispherical radii — this badly overcounts the volume.
Adding the two hemisphere volumes separately instead of combining them as one sphere (arithmetically equivalent but more error-prone).

Real-world relevance

This capsule shape is exactly the geometry of a medicine capsule, an LPG cylinder, and a pressure vessel/submarine hull, since hemispherical ends distribute internal pressure more evenly than flat ends.

Common Mistakes

1Taking the cylinder height as the full 20 cm instead of subtracting the two hemispherical radii (should be 13 cm).
2Using the sphere volume as $\frac{2}{3}\pi r^3$ (one hemisphere) instead of $\frac{4}{3}\pi r^3$ for the two hemispheres combined.
3Rounding r or π too early, or using diameter 7 as the radius, which throws off the whole computation.

Interesting Facts

A cylinder with hemispherical ends is exactly the shape of a medicine capsule, an LPG gas cylinder and a submarine hull — hemispherical caps spread internal pressure evenly and avoid the stress concentrations that flat ends create.

Two hemispheres of equal radius always join into one complete sphere, so their combined volume is $\frac{4}{3}\pi r^3$ — the same formula Archimedes derived over 2,200 years ago.

For this capsule the sphere (two caps) makes up about 26% of the total volume while the 13-cm cylinder body makes up about 74%, showing how much the central tube dominates.

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Frequently Asked Questions

Why is the cylinder height 13 cm and not 20 cm?

The total height of 20 cm includes the two hemispherical caps. Each cap adds a length equal to the radius (3.5 cm), so together they add 7 cm. Subtracting these gives the cylindrical portion: $20 - 7 = 13\text{ cm}$ .

Why do the two hemispheres form a sphere?

Both hemispheres have the same radius as the cylinder (3.5 cm). Two hemispheres of equal radius fit together to make one complete sphere, so their combined volume is the sphere formula $\frac{4}{3}\pi r^3$ instead of adding two separate $\frac{2}{3}\pi r^3$ terms.

Is the answer 680.17 cm³ exact?

The exact value is $4081/6\text{ cm}^3$ , which equals $680.1666...\text{ cm}^3$ , usually written as $680.17\text{ cm}^3$ or $680\frac{1}{6}\text{ cm}^3$ . Any of these forms is acceptable in the exam as long as the working is shown.