A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

Understand the Question When a hemispherical depression is cut out from one face of a cubical block, the inner curved surface of the hemisphere gets exposed while the circular base on the top face is removed. Surface Area of Remaining Solid Edge of the cube , and radius of the hemisphere . Step 1 · Identify Dimensions of the Cube and Hemisphere Given edge of the cube and diameter of the hemisphere . Radius of the hemisphere: Step 2 · Calculate Surface Area of the Remaining Solid Substitute Answer > Common Mistakes Subtracting CSA Instead of Adding: Confusing volume with surface area. Carving out a depression hollows out space, which adds the curved surface area of the hemisphere to the total exposed surface area. Forgetting the Base Area: Failing to subtract the circular top area () from the top face of the cube. Using Diameter as Radius: Using instead of in the formulas.

Surface Areas and Volumes | Exercise 12.1

Question 5

A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter $l$ of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

Solution

Understand the Question

When a hemispherical depression is cut out from one face of a cubical block, the inner curved surface of the hemisphere gets exposed while the circular base on the top face is removed.
Surface Area of Remaining Solid $= \text{TSA of Cube} + \text{CSA of Hemisphere} - \text{Base Area of Hemisphere}$
Edge of the cube $= l$ , and radius of the hemisphere $= r = \dfrac{l}{2}$ .

Step 1 · Identify Dimensions of the Cube and Hemisphere

Given edge of the cube $= l$ and diameter of the hemisphere $= l$ . Diagram 1

Radius of the hemisphere: $r = \dfrac{l}{2}$

Step 2 · Calculate Surface Area of the Remaining Solid

$\text{Surface area of remaining solid} = \text{TSA of cube} + \text{CSA of hemisphere} - \text{Area of base of hemisphere}$

\begin{aligned} &= 6l^2 + 2\pi r^2 - \pi r^2 \\[0.6em] &= 6l^2 + \pi r^2 \end{aligned}

Substitute $r = \dfrac{l}{2}$

\begin{aligned} &= 6l^2 + \pi \left(\dfrac{l}{2}\right)^2 \\[0.6em] &= 6l^2 + \pi \dfrac{l^2}{4} \\[0.6em] &= \left(6 + \dfrac{\pi}{4}\right)l^2 = \dfrac{l^2}{4}(24 + \pi) \end{aligned}

Answer

$\dfrac{1}{4}l^2(24 + \pi)$ \text{ or } $\left(6 + \dfrac{\pi}{4}\right)l^2$

Common Mistakes

Subtracting CSA Instead of Adding: Confusing volume with surface area. Carving out a depression hollows out space, which adds the curved surface area of the hemisphere to the total exposed surface area.
Forgetting the Base Area: Failing to subtract the circular top area ( $\pi r^2$ ) from the top face of the cube.
Using Diameter as Radius: Using $r = l$ instead of $r = \dfrac{l}{2}$ in the formulas.

Question 5

More questions in Exercise 12.1