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# Quantitative Aptitude :: Volume and Surface Area

## Volume and Surface Area Important Formulas

1. CUBOID:

A cuboid is a 3D shape. Cuboids have six faces, which form a convex polyhedron. Broadly, the faces of the cuboid can be any quadrilateral. More narrowly, cuboids are made from 6 rectangles, which are placed at right angles. A cuboid that uses all square faces is a cube.

Let length = l, breadth = b and height = h units. Then

$$\mathbb{i.}$$ Volume = (l x b x h) cubic units.

$$\mathbb{ii.}$$ Surface area = 2(lb + bh + lh) sq. units.

$$\mathbb{iii.}$$ Diagonal = $$\sqrt{l^{2} \ + \ b^{2} \ + \ h^{2}}$$ units.

2. CUBE:

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

Let each edge of a cube be of length a. Then,

$$\mathbb{i.}$$ Volume = a3 cubic units.

$$\mathbb{ii.}$$ Surface area = 6a2 sq. units.

$$\mathbb{iii.}$$ Diagonal = $$\sqrt{3}a$$ units.

4. CONE:

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.

Let radius of base = r and Height (or length) = h. Then,

$$\mathbb{i.}$$ Slant height $$l \ = \ \sqrt{h^{2} \ + \ r^{2}}$$ units.

$$\mathbb{ii.}$$ Volume = r$$\left( \frac{1}{3} \ pi r^{2}h \right)$$ cubic units.

$$\mathbb{iii.}$$ Curved surface area = $$(\pi rl)$$ sq. units.

$$\mathbb{iv.}$$ Total surface area = $$(\pi rl \ + \ \pi r^{2})$$ sq. units.

4. CYLINDER:

A cylinder is one of the most basic curved geometric shapes, with the surface formed by the points at a fixed distance from a given line segment, known as the axis of the cylinder

Let radius of base = r and Height (or length) = h. Then,

$$\mathbb{i.}$$ Volume = $$\left(\pi r^{2}h \right)$$ cubic units.

$$\mathbb{ii.}$$ Curved surface area = $$2(\pi rh)$$ sq. units.

$$\mathbb{iii.}$$ Total surface area = $$2\pi r(h \ + \ r)$$ sq. units.

5. SPHERE:

Sphere, In geometry, the set of all points in three-dimensional space lying the same distance (the radius) from a given point (the centre), or the result of rotating a circle about one of its diameters.

Let the radius of the sphere be r. Then,

$$\mathbb{i.}$$ Volume = $$\left( \frac{4}{3} \ pi r^{3} \right)$$ cubic units.

$$\mathbb{ii.}$$ surface area = $$(4\pi r^{2})$$ sq. units.

6. HEMISPHERE:

A hemisphere is the half of a sphere. When a sphere is cut into two halves, then the shape we get is called the hemisphere. A hemisphere has a curved surface and a flat base. The curved surface area of the hemisphere is half of the surface area of the sphere.

Let the radius of a hemisphere be r. Then,

$$\mathbb{i.}$$ Volume = $$\left( \frac{2}{3} \ pi r^{3} \right)$$ cubic units.

$$\mathbb{ii.}$$ Curved surface area = $$(2\pi r^{2})$$ sq. units.

$$\mathbb{iv.}$$ Total surface area = $$(3\pi r^{2})$$ sq. units.

Note: 1 litre = 1000 cm3.