# Right Pyramid

A right pyramid has its apex aligned right above the center of the base.

## Definition

A right pyramid is a pyramid whose apex is aligned exactly above the center of the base. Thus, a perpendicular line joining the apex and the base center is the height of a right pyramid.

Unlike a right pyramid, an oblique pyramid does not have its apex aligned away from the center of the base.

Some common right pyramids with different bases are:

Right Triangular Pyramid

It has a triangular base with its apex aligned directly above its center.

Right Square Pyramid

It has a square base with its apex aligned directly above its center.

Right Rectangular Pyramid

It has a rectangular base with its apex aligned directly above its center.

Right Pentagonal Pyramid

It has a pentagonal base with its apex aligned directly above its center.

Right Hexagonal Pyramid

It has a hexagonal base with its apex aligned directly above its center.

Like all other polyhedrons, we can calculate the surface area and volume of a right pyramid.

## Formulas

### Surface Area

The formulas is:

Surface Area (SA) = ${B+\dfrac{1}{2}Ps}$, here B = base area, P = base perimeter, s = slant height,

Also ${\dfrac{1}{2}Ps}$ = lateral surface area (LSA)

∴ SA = B + LSA

Let us solve some examples to understand the above concept better.

Find the lateral and the total surface area of a right pyramid with a base perimeter of 36 cm, base area of 81 cm2, and slant height of 16 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = ${\dfrac{1}{2}Ps}$, here P = 36 cm, s = 16 cm
LSA  = ${\dfrac{1}{2}\times 36\times 16}$
= 288 cm2
Total Surface Area (TSA) = B + LSA, here B = 81 cm2, LSA = 288 cm2
TSA  = 81 + 288
= 369 cm2

Find the surface area of a right pyramid with a square base of 6 cm, and a slant height of 9.5 cm.

Solution:

Here we will use the specific surface area formula for a right square pyramid.
Total Surface Area (TSA) = b2 + 2bs, here b = 6 cm, s = 9.5 cm
TSA  = 62 + 2 × 6 × 9.5
= 150 cm2

### Volume

It is the space it occupies in a 3-dimensional plane. It is expressed in cubic units such as m3, cm3, mm3, and in3.

The general formula is given below:

Volume (V) = ${\dfrac{1}{3}Bh}$, here B = base area, h = height

Let us solve an example to understand the above concept better.

Find the volume of a right pyramid with a base area of 160 cm2 and a height of 16 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{1}{3}Bh}$, here B = 160 cm2, h= 16 cm
V  = ${\dfrac{1}{3}\times 160\times 16}$
= 853.33 cm3

Find the volume of a right triangular pyramid with a basearea of 2.3 cm2 and a height of 5 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{1}{3}Bh}$, here B = 2.3 cm2, h = 5 cm
V  = ${\dfrac{1}{3}\times 2.3\times 5}$
= 3.83 cm3

Find the volume of a right square pyramid with a base of 6 cm and a height of 8 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{1}{3}b^{2}h}$, here b = 6 cm, h = 8 cm
V  = ${\dfrac{1}{3}\times 6^{2}\times 8}$
= 96 cm3

Find the volume of a right rectangular pyramid with a base length of 10 cm, a base width of 6 cm, and a height of 12 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{1}{3}lwh}$, here l = 10 cm, w = 6 cm, h = 12 cm
V = ${\dfrac{1}{3}\times 10\times 6\times 12}$
= 240 cm3

1. Nemiks says: