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Ordering Mixed Fractions & Negative Fractions Coterminal Angles, Initial/Terminal side Volume of a Cone Angles inside a Circle Factors of Numbers, Perfect Numbers Volume of a Cone MeanA Pyramid is a solid figure with a flat shape forming the base, along with a certain number of
triangles which make up the lateral faces or sides of the pyramid.

The lateral faces meet at a common point above the base, which is the top/apex of the Pyramid.

Pyramids are named depending on the shape of the base.

In fact, another name for a triangular pyramid is a tetrahedron.

If all the sides of the base of a pyramid are the same length, the pyramid is a **regular
pyramid**.

If all the sides of the base are NOT all the same length, then the pyramid is classed as an **irregular
pyramid**.

Depending on where the top/apex of a pyramid is located, the pyramid can be either "regular" or
"oblique".

The top of a regular pyramid, is directly above the center of the base.

Where as the top of an oblique pyramid, is NOT directly above the center of the base.

Working out the surface area of an oblique pyramid usually takes more a bit more time and effort to establish, rather than a right pyramid.

The slant height in a Pyramid is different to the vertical perpendicular height from the base to the
top.

The slant height is the length from the bottom of one of the faces, to the top.

In a right regular pyramid, the slant height is the same on each face/side.

Opening up a right regular square Pyramid, and laying all sides flat out.

**s** is the slant height of the actual 3D pyramid.

There are actually  2 parts that make up the total surface area of a pyramid.

One part is the area of the base.

The second part, is the lateral area, which is the area of all the faces/sides added up together.

So the total surface area of a right regular square pyramid, is made up of 1 square base, and 4
triangle faces/sides that are of equal size.

Base Area =

Area of 1 triangle face = \bf{\frac{1}{2}} ×

Thus,

Lateral Area =

=> Total Pyramid Area =

What is the surface area of the following square pyramid?

Area =

Area is **80cm ^{2}**.

What is the surface area of the following square pyramid?

We don't initially have the slant height of the pyramid, but it can be obtained with Pythagoras.

If we can picture looking at the pyramid directly, as if at eye level.

Now:

Area =

Area is

What is the surface area of the following square pyramid?

This example is a little different, a rectangular base where all sides are NOT the same length, makes
this an irregular pyramid.

A few more sums are needed to come up with a value for the total surface area of a pyramid here.

There is enough information to work out the slant heights. Looking at each side of the pyramid head
on, pythagoras theorem can be used.

The faces at **A** and **C** will share the same slant height, as will the faces at sides
**B** and **D**.

The rectangle base area is given by **4** × **6** = **24**.

Now total surface are is given by

As some decimal values were rounded, this isn't an exact value for the surface area.

But we're close enough to say that the total surface area of this rectangular pyramid is roughly
**79.66m ^{2}**.

In fact, from the calculations done here it can be deduced that for any right rectangular pyramid of the form:

The surface area amounts to:

Area =

As **a**×**s1** and **b**×**s2** in the example above, were multiplied by half,
then multiplied by **2**.

For right regular pyramids, where the base is a regular polygon, there is a more general formula for
finding the surface area of a pyramid.

Which is ( **BASE AREA** ) + ( \bf{\frac{1}{2}} × **perimeter**
× **slant height** ).

This is shown in more detail on a separate page, which can be seen ** here**.

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