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: 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) 3πd² — diameter $2d$ gives radius $r = d$; TSA of solid hemisphere = $3\pi r^2 = 3\pi d^2$.

Q: What is the formula for total surface area of a solid hemisphere?

$\text{TSA} = 3\pi r^2$, which is the sum of the curved surface area ($2\pi r^2$) and the area of the flat circular base (πr²).

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) 3πd² — diameter $2d$ gives radius $r = d$ ; TSA of solid hemisphere = $3\pi r^2 = 3\pi d^2$ .

total surface area of hemispherecurved surface areasolid hemisphereradius from diameter3πr² formulasurface areas and volumes

Marking Scheme

11 mark: correctly selecting option (a) $3\pi d^2$ by correctly identifying $\text{radius} = d$ from diameter = $2d$ and applying $\text{TSA} = 3\pi r^2$ .
2No partial marks for MCQs; working may be shown for verification only.

Hint

TSA of a solid hemisphere = $3\pi r^2$ ; here diameter = $2d$ means radius $r = d$ , so substitute directly.

Quick Oral Answer

TSA of a solid hemisphere is $3\pi r^2$ (curved surface $2\pi r^2$ plus base πr²); since diameter = $2d$ gives $\text{radius} = d$ , $\text{TSA} = 3\pi d^2$ .

Analysis & Explanation

Checks careful reading of the given quantity ' $2d$ ' as the diameter, not the radius.

Concept

Total surface area of a solid hemisphere of radius r is $\text{TSA} = 3\pi r^2$ (curved surface $2\pi r^2$ + flat base πr²).

Key points

Diameter = $2d$ ⟹ radius $r = d$ .
Substituting: $\text{TSA} = 3\pi (d)^2 = 3\pi d^2$ .

Common mistakes

Misreading ' $2d$ ' and taking the radius itself as $2d$ instead of the diameter, leading to wrong distractors like 2πd² or (3/4)πd².
Forgetting to add the base circle's area (πr²) to the curved surface area ( $2\pi r^2$ ).

Real-world

Used to compute the material needed to manufacture solid hemispherical objects such as bowls, domes, and hemispherical ends of storage tanks and silos.

Common Mistakes

1Misreading 'diameter = $2d$ ' and assuming the radius itself equals $2d$ (instead of correctly deriving $\text{radius} = d$ ), leading to the wrong option.
2Using the curved surface area formula ( $2\pi r^2$ ) alone instead of the total surface area formula ( $3\pi r^2$ ), which also includes the flat circular base.
3Confusing total surface area of a hemisphere with that of a full sphere ( $4\pi r^2$ ).

Interesting Facts

The curved surface area of a hemisphere ( $2\pi r^2$ ) is exactly twice the area of its circular base (pi*r squared), a relationship discovered using methods that trace back to Archimedes' work on spheres and cylinders.

Archimedes considered the relationship between a sphere's surface area and its circumscribing cylinder one of his greatest discoveries and reportedly asked for a sphere-and-cylinder diagram to be engraved on his tombstone.

A solid hemisphere's total surface area (curved plus flat circular base) equals $3\pi r^2$ , which is why the coefficient 3 appears in this formula.

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Frequently Asked Questions

What is the formula for total surface area of a solid hemisphere?

$\text{TSA} = 3\pi r^2$ , which is the sum of the curved surface area ( $2\pi r^2$ ) and the area of the flat circular base (πr²).

Why is radius equal to d and not $2d$ here?

Because the question states the diameter is '2d', and diameter = 2 × radius, so $\text{radius} = \frac{2d}{2} = d$ .