Surface Area of Solids

The surface area of a solid is the total area of all its faces (the outside surfaces).

🌟 Surface Area of a Cuboid:

• A cuboid has 6 faces, grouped in 3 pairs.
• Each pair of faces is the same size.

Let’s call:

• Length = $l$
• Width = $w$
• Height = $h$

To find the total surface area:

• The top and bottom faces are identical. Each has an area of $l \times w$.
• The front and back faces are identical. Each has an area of $l \times h$.
• The left and right faces are identical. Each has an area of $w \times h$.

By adding them all together, you get:

🌟 Surface Area of a Cuboid Formula:

$$
\large
\begin{align*}
\text{SA} &= 2lw + 2lh + 2wh \\[0.5em]
&= 2(lw + lh + wh)
\end{align*}
$$

Exam Tip: Remember, every cuboid has 3 different face shapes, and each appears twice:)

A cube has 6 identical square faces.

If each side of the cube is $a$:

• Area of each face:  $a \times a = a^2$
• Total surface area:  6 faces, so $6 \times a^2$

🌟 Surface Area of a Cube Formula:

$$ \large \text{SA} = 6a^2$$

Exam Tip: Every face on a cube is a square, and all 6 faces are same size:)

A cuboid has:

• Length ($l$) = 10 cm
• Width ($w$) = 5 cm
• Height ($h$) = 8 cm

To find the surface area, calculate the area of each pair of faces and add them up:

1. Top + Bottom: $ \quad 2 \times l \times w = 2 \times 10\,\text{cm} \times 5\,\text{cm} = 100\,\text{cm}^2 $
2. Front + Back: $ \quad 2 \times l \times h = 2 \times 10\,\text{cm} \times 8\,\text{cm} = 160\,\text{cm}^2$
3. Left + Right: $ \quad 2 \times w \times h = 2 \times 5\,\text{cm} \times 8\,\text{cm} = 80\,\text{cm}^2$

Adding them together:

$$ \text{Surface Area} = 100 \text{ cm}^2 + 160\text{ cm}^2 + 80\text{ cm}^2 = 340\text{ cm}^2$$

This triangular prism has 5 faces:

• 2 identical triangles (front and back)
• 3 rectangles (base, top, and side)

Let’s find the surface area of this prism step by step:

• Front triangle: $\quad \frac{1}{2} \times 4\text{ cm} \times 3\text{ cm} = 6\text{ cm}^2$

• Back triangle: $\quad \frac{1}{2} \times 4\text{ cm} \times 3\text{ cm} = 6\text{ cm}^2$

• Base rectangle: $\quad 4\text{ cm} \times 7\text{ cm} = 28\text{ cm}^2$

• Top rectangle: $\quad 7\text{ cm} \times 5\text{ cm} = 35\text{ cm}^2$

• Side rectangle: $\quad 7\text{ cm} \times 3\text{ cm} = 21\text{ cm}^2$

Add all of them together:

$$ \text{Surface Area} = 6\text{ cm}^2 + 6\text{ cm}^2 + 28\text{ cm}^2 + 35\text{ cm}^2 + 21\text{ cm}^2 = 96\text{ cm}^2$$