Mensuration - Surface Area of Cube, Cuboid, and Cylinder

A closed rectangular box has a length of $12 cm$, breadth of $10 cm$, and height of $8 cm$. Calculate the total surface area of the box.

Given: $l = 12 cm$, $b = 10 cm$, $h = 8 cm$. Using the formula for Total Surface Area (TSA) of a cuboid: $TSA = 2(lb + bh + hl)$ $TSA = 2(12 \times 10 + 10 \times 8 + 8 \times 12)$ $TSA = 2(120 + 80 + 96)$ $TSA = 2(296)$ $TSA = 592 cm^2$

To find the TSA of a cuboid, we calculate the area of the three distinct pairs of rectangular faces and sum them up. The calculation shows the area of the top/bottom ($120 cm^2$), the sides ($80 cm^2$), and the front/back ($96 cm^2$), all multiplied by 2.

Find the curved surface area and total surface area of a cylinder with a base radius of $7 cm$ and a height of $15 cm$. (Take $\pi = \frac{22}{7}$)

Given: $r = 7 cm$, $h = 15 cm$.

1. Curved Surface Area (CSA): $CSA = 2\pi rh$ $CSA = 2 \times \frac{22}{7} \times 7 \times 15$ $CSA = 44 \times 15 = 660 cm^2$
2. Total Surface Area (TSA): $TSA = 2\pi r(r + h)$ $TSA = 2 \times \frac{22}{7} \times 7 \times (7 + 15)$ $TSA = 44 \times 22 = 968 cm^2$

The CSA represents the side 'label' area of the cylinder. The TSA adds the area of the two circular lids ($2 \times \pi r^2$) to the CSA. By using the distributive property formula $2\pi r(r+h)$, the calculation becomes more efficient.

Let the edge of the cube be $a$. Given, $TSA = 600 cm^2$. We know, $TSA = 6a^2$. $6a^2 = 600$ $a^2 = \frac{600}{6}$ $a^2 = 100$ $a = \sqrt{100}$ $a = 10 cm$ Thus, the edge of the cube is $10 cm$.

To find the edge when the surface area is given, we equate the given value to the formula $6a^2$ and solve for $a$.

Let $h$ be the height of the cylinder and $r$ be its radius. When the cylinder is cut along its height, the width of the rectangular sheet equals the height of the cylinder, and the length of the sheet equals the circumference of the base. Width of sheet ($h$) = $33 cm$. Area of sheet = Lateral Surface Area of cylinder = $4224 cm^2$. Length of sheet ($l$) $\times$ Width ($h$) = $4224$ $l \times 33 = 4224$ $l = \frac{4224}{33}$ $l = 128 cm$ Perimeter of the rectangular sheet = $2(l + h)$ $P = 2(128 + 33)$ $P = 2(161)$ $P = 322 cm$ The perimeter of the sheet is $322 cm$.

When a cylinder is opened into a rectangle, the circumference of the circle becomes the length of the rectangle and the height of the cylinder becomes the width.