Volume and Surface Formulas

Volume expresses the quantity of something(for example water) we need to fill in a shape. Space figures have volume only. Plane figures(triangles, squares) do not have volume.
The standard notation for volume is V.

Rectangular parallelepiped

rectangular parallelepiped

A rectangular parallelepiped has 6 faces that are rectangles.
If the sides of the rectangle at the bottom are a and b and the height of the parallelepiped is c (the third edge of the rectangular parallelepiped).
The volume formula is:

$V = a \cdot b \cdot c$
Surface area = $2(a \cdot b + a \cdot c + b \cdot c)$


A cube is a rectangular parallelepiped whose faces and edges are all equal.


If an edge of a cube is a then its volume is:

$V = a \cdot a \cdot a = a^3$
Surface area = $6a \cdot a = 6a^2$



A parallelepiped is formed by 6 parallelograms. If the area of the bottom is A and the height of the parallelepiped is h.
The volume formula is:

$V = A \cdot h$



A pyramid is a figure with a polygonal base(triangle, square, rectangle) connected with one point called apex.
Let the height(the distance between the apex and the base) of the pyramid be h and the area of the base is A.
The volume is given by the formula:

$V = \frac{1}{3} \cdot A \cdot h$
Regular Tetrahedron
Regular Tetrahedron
$V = \frac{\sqrt{2}\cdot a^3}{12}$
Surface area $= \sqrt{3}\cdot a^2$

Right circular cone


A cone is a figure with base circle connected with a point called apex or vertex.
If the area of the base is A and the height of the cone is h its volume is:

$V = \frac{1}{3} \cdot A \cdot h = \frac{1}{3} \cdot \pi \cdot r^2\cdot h$
Lateral surface area = $\pi\cdot r \cdot l$
The total surface area of a right circular cone is the lateral surface area + the area of the bottom.
Total surface area = $\pi\cdot r(r + l)$



A sphere is the surface of a completely round ball.
Every sphere has a has a central point of called "center" of the sphere.
Radius is the length from the center to any point on the surface of the sphere.
The volume of the sphere with radius r is:

$V = \frac{4}{3} \cdot \pi \cdot r^3$
Surface area = $4\cdot\pi\cdot r^2$


Circular cylinder

A circular cylinder is figure that has two identical and parallel circular bases.
Let the radius of the bottom be r and the height(the distance between the both bases) of the cylinder be h.
The volume formula is:

$V = \pi \cdot r^2 \cdot h$

Right Circular Cylinder

Right circular cylinder
$V = \pi \cdot r^2 \cdot h$

Area of the curved surface of the cylinder:
Curved(lateral) surface area = $2\cdot\pi\cdot r \cdot h$

Total surface area = curved(lateral) surface area + area of two circular ends:
Total surface area = $2\cdot\pi\cdot r(h + r)$

Quiz: Volume and surface area

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