Q16

1 markMCQSection A

The total surface area of a solid hemisphere of diameter '2d' is : (a) $3\pi d^2$ (b) $2\pi d^2$ (c) $\frac{1}{2}\pi d^2$ (d) $\frac{3}{4}\pi d^2$

Q: 16. The total surface area of a solid hemisphere of diameter '2d' is : (a) $3\pi d^2$ (b) $2\pi d^2$ (c) $\frac{1}{2}\pi d^2$ (d) $\frac{3}{4}\pi d^2$

(a) $3\pi d^2$, since radius $r = d$, and TSA of a solid hemisphere = $2\pi r^2 + \pi r^2 = 3\pi r^2$ = $3\pi d^2$.

Q: Why is the TSA of a solid hemisphere 3πr² and not 2πr²?

A solid hemisphere has a curved surface ($2\pi r^2$) plus a flat circular base ($\pi r^2$). Adding them gives $3\pi r^2$. Only a hollow hemispherical bowl uses $2\pi r^2$.

Q: How do we get r = d from diameter 2d?

Radius is half the diameter, so $r = \frac{2d}{2} = d$. Substituting this into $3\pi r^2$ gives $3\pi d^2$.

Q: How is this different from a full sphere?

A full sphere's surface area is $4\pi r^2$. A solid hemisphere is $3\pi r^2$ — three-quarters of the sphere's surface, because of its added flat base.

Surface Areas and Volumes

Total surface area of a solid hemisphere

Options

(A)

3\pi d^2

(B)

2\pi d^2

(C)

\frac{1}{2}\pi d^2

(D)

\frac{3}{4}\pi d^2

Official Answer

(a) $3\pi d^2$ , since radius $r = d$ , and TSA of a solid hemisphere = $2\pi r^2 + \pi r^2 = 3\pi r^2$ = $3\pi d^2$ .

total surface areasolid hemisphere3πr²curved surface area 2πr²flat base πr²radius = ddiameter 2d3πd²

Marking Scheme

11 mark for correct option (a) $3\pi d^2$ .
2Expected reasoning: $r = \frac{2d}{2} = d$ ; $\text{TSA} = 2\pi r^2 + \pi r^2 = 3\pi r^2 = 3\pi d^2$ .
3Award only if the flat base is included ( $3\pi r^2$ , not $2\pi r^2$ ).

Hint

TSA of a SOLID hemisphere = $3\pi r^2$ (curved 2πr² + flat base πr²). First convert diameter 2d to radius $r = d$ .

Quick Oral Answer

For a solid hemisphere the total surface area is curved plus base, that is 2 pi r squared plus pi r squared, equal to 3 pi r squared; with radius d it becomes 3 pi d squared.

Analysis & Explanation

Tests the formula for total surface area of a solid hemisphere, including the flat circular base.

Concept

TSA of a solid hemisphere = curved surface area + area of flat base = $2\pi r^2 + \pi r^2 = 3\pi r^2$ .
Diameter given is 2d, so radius $r = \frac{2d}{2} = d$ .
Substituting r = d gives $\text{TSA} = 3\pi d^2$ .

Common mistakes

Using only the curved surface area ( $2\pi r^2$ ) and forgetting the flat circular base, since the solid stands directly on it.
Misreading the diameter as 'd' instead of '2d', which would give a wrong radius.

Common Mistakes

1Giving $2\pi d^2$ by using only the curved surface area and forgetting the flat circular base of a solid hemisphere.
2Not converting diameter 2d to radius d, leading to fractions like $\frac{3}{4}\pi d^2$ .
3Confusing hemisphere TSA ( $3\pi r^2$ ) with full sphere surface area ( $4\pi r^2$ ).

Interesting Facts

A solid hemisphere's total surface area ( $3\pi r^2$ ) is exactly three-quarters of a full sphere's surface area ( $4\pi r^2$ ), even though it is only half the solid.

The extra flat circular base is what distinguishes a 'solid' hemisphere from a 'hollow' hemispherical shell in CBSE problems — a single word changes the formula from $3\pi r^2$ to $2\pi r^2$ .

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Frequently Asked Questions

Why is the TSA of a solid hemisphere 3πr² and not 2πr²?

A solid hemisphere has a curved surface ( $2\pi r^2$ ) plus a flat circular base ( $\pi r^2$ ). Adding them gives $3\pi r^2$ . Only a hollow hemispherical bowl uses $2\pi r^2$ .

How do we get r = d from diameter 2d?

Radius is half the diameter, so $r = \frac{2d}{2} = d$ . Substituting this into $3\pi r^2$ gives $3\pi d^2$ .

How is this different from a full sphere?

A full sphere's surface area is $4\pi r^2$ . A solid hemisphere is $3\pi r^2$ — three-quarters of the sphere's surface, because of its added flat base.