Volume and surface area of a hemisphere of radius 7cm

Question

Find the volume and total surface area of a solid hemisphere of radius 7 cm. (Take $\pi = 22/7$ )

Solution — Step by Step

A hemisphere is exactly half a sphere.

For a sphere of radius $r$ :

Volume = $\frac{4}{3}\pi r^3$
Surface area = $4\pi r^2$

For a solid hemisphere of radius $r$ :

Volume = $\frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$
Curved surface area (CSA) = $\frac{1}{2} \times 4\pi r^2 = 2\pi r^2$ (the dome part)
Total surface area (TSA) = curved surface area + circular base = $2\pi r^2 + \pi r^2 = 3\pi r^2$

The total surface area includes the flat circular base, which has area $\pi r^2$ .

Given: $r = 7$ cm, $\pi = 22/7$ .

V = \frac{2}{3}\pi r^3 = \frac{2}{3} \times \frac{22}{7} \times 7^3

= \frac{2}{3} \times \frac{22}{7} \times 343

= \frac{2}{3} \times 22 \times 49 \quad \left(\text{since } \frac{343}{7} = 49\right)

= \frac{2 \times 22 \times 49}{3} = \frac{2156}{3} = 718.\overline{6} \text{ cm}^3

\boxed{V = \frac{2156}{3} \approx 718.67 \text{ cm}^3}

\text{TSA} = 3\pi r^2 = 3 \times \frac{22}{7} \times 7^2 = 3 \times \frac{22}{7} \times 49

= 3 \times 22 \times 7 = 3 \times 154 = 462 \text{ cm}^2

\boxed{\text{TSA} = 462 \text{ cm}^2}

Why This Works

When we cut a sphere in half, we get two identical hemispheres. The volume is exactly half the sphere volume. The dome-shaped surface (curved surface area) is half the sphere’s surface area. But we also now have a new flat circular face — the cross-section — which adds $\pi r^2$ to the surface area.

The total surface area being $3\pi r^2$ makes sense: it’s $2\pi r^2$ (dome) + $1\pi r^2$ (flat base) = $3\pi r^2$ .

If the hemisphere is hollow (like a bowl), you would only use the curved surface area $2\pi r^2$ without the base. The choice of CSA vs TSA depends on what the problem is asking — read the question carefully.

Alternative Method

Direct calculation without the sphere formula: recognise that the curved surface is exactly half a sphere, which has surface area $4\pi r^2$ . So CSA = $2\pi r^2$ . Then add the base $\pi r^2$ to get TSA = $3\pi r^2$ .

Common Mistake

The most frequent error is using TSA = $2\pi r^2$ (forgetting the base) when the question asks for the TOTAL surface area of a SOLID hemisphere. The curved surface area is $2\pi r^2$ , but the total surface area includes the flat circular base: TSA = $2\pi r^2 + \pi r^2 = 3\pi r^2$ . If the question says “open hemisphere” (like a bowl), then use CSA = $2\pi r^2$ only.

When $r = 7$ cm and $\pi = 22/7$ , many terms simplify nicely because 7 cancels with the denominator of $\pi$ . Always look for this cancellation before calculating — it saves time and reduces arithmetic errors.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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