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Mensuration - Surface Area of Cube, Cuboid, and Cylinder

Grade 8CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Surface Area is the measure of the total area that the surface of a three-dimensional object occupies. Imagine peeling the skin of an orange and laying it flat; the area of that flat skin is the surface area.

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A Cuboid is a solid shape with six rectangular faces. It is defined by its length (ll), breadth (bb), and height (hh). Visually, it consists of three pairs of identical opposite faces: top-bottom, front-back, and two side faces.

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A Cube is a specific type of cuboid where all dimensions are equal (l=b=h=al = b = h = a). Every face of a cube is a square of the same size. If you were to unfold a cube, you would see a net of six identical squares connected together.

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Lateral Surface Area (LSA) refers to the area of all the faces of a solid excluding the area of its top and bottom bases. For example, the LSA of a room represents the total area of the four walls, excluding the floor and the ceiling.

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A Cylinder is a solid with two congruent circular bases and a curved lateral surface. If you imagine a cylinder as a soup can, the label wrapped around the side represents the Curved Surface Area (CSA).

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The Curved Surface Area of a cylinder can be visualized by cutting the curved surface vertically and rolling it out flat. This flat shape is a rectangle whose length is equal to the circumference of the circle (2πr2\pi r) and whose width is the height (hh) of the cylinder.

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Total Surface Area (TSA) of a cylinder is the sum of the Curved Surface Area and the areas of the two circular bases (top and bottom). Visually, this includes the rectangular 'wrap' and the two circular 'lids'.

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Units of Measurement: Since surface area is a two-dimensional measure of a three-dimensional object, it is always expressed in square units, such as cm2cm^2, m2m^2, or mm2mm^2.

📐Formulae

Total Surface Area of a Cuboid = 2(lb+bh+hl)2(lb + bh + hl)

Lateral Surface Area of a Cuboid = 2h(l+b)2h(l + b)

Total Surface Area of a Cube = 6a26a^2

Lateral Surface Area of a Cube = 4a24a^2

Curved Surface Area (CSA) of a Cylinder = 2πrh2\pi rh

Total Surface Area (TSA) of a Cylinder = 2πr(r+h)2\pi r(r + h)

Area of one circular base of a Cylinder = πr2\pi r^2

💡Examples

Problem 1:

Find the total surface area of a cuboid whose length is 12 cm12\text{ cm}, breadth is 8 cm8\text{ cm}, and height is 5 cm5\text{ cm}.

Solution:

Given: l=12 cml = 12\text{ cm}, b=8 cmb = 8\text{ cm}, h=5 cmh = 5\text{ cm}.\Using the formula for Total Surface Area (TSA) of a cuboid:\TSA=2(lb+bh+hl)TSA = 2(lb + bh + hl)\TSA=2(12×8+8×5+5×12)TSA = 2(12 \times 8 + 8 \times 5 + 5 \times 12)\TSA=2(96+40+60)TSA = 2(96 + 40 + 60)\TSA=2(196)TSA = 2(196)\TSA=392 cm2TSA = 392\text{ cm}^2.

Explanation:

To find the total surface area, we calculate the area of all six rectangular faces by summing the products of the dimensions and doubling the result because opposite faces are equal.

Problem 2:

A cylindrical tank has a radius of 7 m7\text{ m} and a height of 3 m3\text{ m}. Find its total surface area. (Take π=227\pi = \frac{22}{7})

Solution:

Given: r=7 mr = 7\text{ m}, h=3 mh = 3\text{ m}.\Using the formula for Total Surface Area (TSA) of a cylinder:\TSA=2πr(r+h)TSA = 2\pi r(r + h)\TSA=2×227×7×(7+3)TSA = 2 \times \frac{22}{7} \times 7 \times (7 + 3)\TSA=2×22×10TSA = 2 \times 22 \times 10\TSA=440 m2TSA = 440\text{ m}^2.

Explanation:

The total surface area includes the curved side of the tank plus the area of the circular top and bottom. We substitute the radius and height into the formula and simplify.