A square pyramid is a pyramid with a square base bounded by four lateral faces meeting at a common point, known as the apex. The lateral faces are triangular.
Square Pyramid
How many faces,edges and vertices does a square pyramid have?
A square pyramid has 5 faces, 8 edges, and 5 vertices.
A net for a square pyramid can illustrate its shape from a 2-D view. This net can be folded along the dotted lines to form a square pyramid as shown in the diagram.
Square Pyramid Net
Types
Based on the position of its apex, a square pyramid can be classified into 2 types – (1) Right square pyramid, and (2) Oblique square pyramid.
Right and Oblique
Right Square Pyramid
A right square pyramid is a pyramid whose apex is aligned right above its base center. So, an imaginary line drawn from the apex intersects the base at its center at a right angle. A square pyramid is usually right.
In contrast, when the apex is away from the base center, the pyramid is an oblique square pyramid. Its height is a perpendicular line from the apex to the base.
A square pyramid has a special type based on its edge lengths.
Equilateral
An equilateral square pyramid has its 3 lateral faces that are equilateral triangles. It means all the edges of are of equal length.
Like all other polyhedrons, we can calculate the surface area and the volume of a square pyramid.
Formulas
Volume
The formula is:
Volume (V)= ${\dfrac{1}{3}b^{2}h}$, here b = base, h = height
Let us solve some examples to understand the concept better.
Find the volume of a square pyramid with a base of 12 cm, and a height of 6 cm.
Solution:
As we know, Volume (V) = ${\dfrac{1}{3}b^{2}h}$, here b = 12 cm, h = 6 cm ∴ V = ${\dfrac{1}{3}\times 12^{2}\times 6}$ = 288 cm3
Find the volume of a square based pyramid with a base of 4 cm, and a height of 14 cm.
Solution:
As we know, Volume (V) = ${\dfrac{1}{3}b^{2}h}$, here b = 4 cm, h = 14 cm ∴ V = ${\dfrac{1}{3}\times 4^{2}\times 14}$, = 74.66 cm3
Surface Area
The formula is:
Surface Area (SA) = ${b^{2}+2bs}$, here b = base, s = slant height
Also ${2bs}$ = lateral surface area (LSA)
∴ SA = b2+ LSA
Let us solve some examples to understand the concept better.
Find the lateral and total surface area of a square pyramid with a slant height of 8.5 cm and a base of 8 cm.
Solution:
As we know, Lateral Surface Area (LSA) = ${2bs}$, here b = 8 cm, s = 8.5 cm ∴ LSA = 2 × 8 × 8.5 = 136 cm2 Total Surface Area (TSA) = b2 + LSA, here b = 8 cm, LSA = 136 cm2 ∴ TSA = 82+ 136 = 200 cm2
Find the surface area of a square based pyramid with a base of 11 cm and a slant height of 15 cm.
Solution:
As we know, Total Surface Area (TSA) = b2 + 2bs, here b = 11 cm, s = 15 cm ∴ TSA = 112+ 2 × 11 × 15 = 451 cm2
Now, let us learn how to find the slant height and height of a square pyramid with some typical examples.
Finding the slant height of a square pyramid when its BASE and SURFACE AREA are known.
Find the slant height of a square pyramid with a base area of 189 cm2 and a base of 9 cm.
Solution:
As we know, Surface Area (SA) = b2 + 2bs ∴ s = ${\dfrac{SA-b^{2}}{2b}}$, here SA = 189 cm2, b = 9 cm = ${\dfrac{189-9^{2}}{2\times 9}}$ = 6 cm
Finding the height of a square pyramid when its BASE and VOLUME are known.
Find the height of a square pyramid with a volume of 256 cm3, and a base of 8 cm.
Solution:
As we know, Volume (V) = ${\dfrac{1}{3}b^{2}h}$ ∴ h = ${\dfrac{3V}{b^{2}}}$, here V = 256 cm3, b = 8 cm = ${\dfrac{3\times 256}{8^{2}}}$ = 12 cm
