How To Find The Slant Height Of A Square Pyramid

Surface Area of a Square Pyramid

In this section, nosotros will learn almost the surface expanse of a foursquare pyramid. A pyramid is a 3-D object whose all side faces are coinciding triangles and whereas its base of operations can be any polygon. Ane side of each of these triangles coincides with one side of the base polygon. A foursquare pyramid is a pyramid whose base is a foursquare. The pyramids are named according to the shape of their bases. Just like other three-dimensional shapes, a square pyramid also has two types of areas.

Total Surface Surface area (TSA)
Lateral Surface Area (LSA)

Let us learn about the surface area of a foursquare pyramid along with the formula and a few solved examples here. You can observe a few practice questions in the stop.

1.	What is the Surface Area of a Square Pyramid?
2.	Formula of Surface Expanse of a Square Pyramid
3.	How to Summate Surface Expanse of Square Pyramid?
4.	FAQs on Surface Area of Square Pyramid

What is the Surface Area of a Square Pyramid?

The discussion "surface" means " the exterior or exterior part of an object or body". So, the total surface area of a foursquare pyramid is the sum of the areas of its lateral faces and its base. We know that a foursquare pyramid has:

a base which is a foursquare.
four side faces, each of which is a triangle.

All these triangles are isosceles and congruent, each of which has a side that coincides with a side of the base of operations (square).

So, the surface area of a square pyramid is the sum of the areas of four of its triangular side faces and the base area which is foursquare.

Formula of Surface Area of a Square Pyramid

Permit u.s. consider a square pyramid whose base'due south length (square's side length) is 'a' and the height of each side face up (triangle) is 'l' (this is also known as the slant peak). i.e., the base and height of each of the 4 triangular faces are 'a' and 'l' respectively. So the base of operations area of the pyramid which is a square is a × a = a^two and the surface area of each such triangular confront is 1/2 × a × l. So the sum of areas of all 4 triangular faces is iv ( ½ al) = 2 al. Allow u.s.a. at present understand the formulas to summate the lateral and total surface area of a square pyramid using height and slant height.

Total Surface Area of Square Pyramid Using Slant Superlative

The total surface area of a square pyramid is the full surface area covered by the iv triangular faces and a square base. The total surface expanse of a foursquare pyramid using slant height can be given by the formula,
Surface area of a square pyramid = a^two+ 2al
where,

a = base of operations length of foursquare pyramid
l = camber peak or height of each side face

Total Surface Area of a Foursquare Pyramid Using Pinnacle

Let united states assume that the height of the pyramid (altitude) be 'h'. Then by applying Pythagoras theorem (you can refer to the below figure),

\(l = \sqrt{\dfrac{a^{2}}{4}+h^{two}}\)

Substituting this in the in a higher place formula,

The surface area of a square pyramid = aⁱⁱ+ 2al = a^two+ 2a\(\sqrt{\dfrac{a^{ii}}{4}+h^{two}}\)

Note: \(\sqrt{\dfrac{a^{2}}{four}+h^{2}}\) can be simplified equally \(\dfrac one 2 \sqrt{a^2+4h^2}\). Thus, the formula of surface area of a foursquare pyramid can be written as a²+ 2a \(\left(\dfrac i 2 \sqrt{a^ii+4h^two}\right)\) = aⁱⁱ+ a\( \sqrt{a^2+4h^two}\).

Lateral Surface Expanse of a Square Pyramid

The lateral surface expanse of a square pyramid is the surface area covered by the four triangular faces. The lateral surface area of a square pyramid using camber height tin be given past the formula,
Lateral surface area of a square pyramid = two al
or,
Lateral surface area of a square pyramid = 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\)
where,

a = base length of square pyramid
l = camber summit or meridian of each side face up
h = height of square pyramid

How to Calculate Surface Area of Square Pyramid?

The surface surface area of a square pyramid can be calculated by representing the 3D effigy into a 2D net. After expanding the 3D figure into a 2D net we will become ane square and four triangles.

The post-obit steps are used to summate the surface area of a square pyramid :

To find the area of the square base: a², 'a' is the base length.
To find the expanse of the four triangular faces: The area of the 4 triangular side faces tin be given as: 2al, 'l' is the camber tiptop. If slant height is non given, nosotros tin calculate it using elevation, 'h' and base of operations length every bit, \(fifty = \sqrt{\dfrac{a^{ii}}{iv}+h^{2}}\)
Add together all the areas together for the full expanse of a foursquare pyramid, while the area of 4 triangular faces gives the lateral area of the foursquare pyramid.
Thus, the surface area of a square pyramid is a²+ 2al and lateral surface area as 2al in squared units.

At present, that we have seen the formula and method to calculate the surface expanse of a foursquare pyramid, permit us take a look at a few solved examples to understand it better.

Examples on Surface Expanse of a Square Pyramid

Example 1: Discover the surface area of a square pyramid of slant height 15 units and base length 12 units.

Solution

The base length of the square pyramid is, a = 12 units.

Its slant acme is, 50 = 15 units.

The surface surface area = a² +2al = 12²+2 (12) (xv) = 504 units²

Answer: The surface area of the given foursquare pyramid is 504 unitsⁱⁱ.
Example two: The summit of a square pyramid is 25 units and the base area of a square pyramid is 256 square units. Notice its surface expanse.

Solution

Let the side of the base of operations (square) be 'a' units.

And then it is given that aⁱⁱ = 256 ⇒ a = xvi units.

The summit of the given square pyramid is h = 25 units.

Surface area of square pyramid = a² + 2a \(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\)

= sixteen^two+two (sixteen) \(\sqrt{\dfrac{xvi^{two}}{4}+25^{2}}\) ≈ 1095.96 square units.

Answer: The surface area of the given square pyramid = 1095.96 square units.

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Do Questions on Surface Expanse of a Foursquare Pyramid

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FAQs on Surface Area of a Foursquare Pyramid

What Is the Surface Surface area of the Square Pyramid?

The surface area of a square pyramid is the sum of the areas of all its 4 triangular side faces with the base area of the square pyramid. If a, h, and fifty are the base length, the height of the pyramid, and camber elevation respectively, and so the expanse of the square pyramid = a²+ 2al (or) a²+2a \(\sqrt{\dfrac{a^{two}}{4}+h^{2}}\).

How Practise Yous Observe the Lateral Expanse of a Square Pyramid?

To find the lateral area of a square pyramid, find the area of i side confront (triangle) and multiply it by iv. If a and fifty are the base of operations length and the slant tiptop of a foursquare pyramid, then the lateral area of the foursquare pyramid = 4 (½ × a × l) = 2al.

If h is the height of the pyramid, and so the lateral area = 2a \(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\).

What Is the Expanse of One of the Triangular Faces of a Square Pyramid?

If a and l are the base length and the camber pinnacle of a square pyramid, then the area of one of the 4 triangular side faces is, ½ × a × 50.

How Do You lot Find the Lateral Area and Surface Surface area of a Square Pyramid?

The lateral surface area of a foursquare pyramid is the sum of the areas of the side faces just, whereas the expanse is the lateral area + area of the base. The lateral expanse of a foursquare pyramid = 2al (or) 2a\(\sqrt{\dfrac{a^{ii}}{four}+h^{2}}\).

To get the total surface surface area, we need to add together the area of the base (which is a²) to each of these formulas. The total surface area of a foursquare pyramid = a² + 2al (or) a² + 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\).
where,

a = Length of the base (foursquare)
l = Camber height
h = Height of the pyramid

How To Calculate Surface Area of a Square Pyramid Without Slant Height?

We know, slant height of a square pyramid is given in terms of pinnacle and base length by the formula, \(fifty = \sqrt{\dfrac{a^{2}}{iv}+h^{2}}\). We tin summate the camber peak from the given meridian and base of operations length and employ the formula for surface area of foursquare pyramid as,
LSA of pyramid = 2a\(\sqrt{\dfrac{a^{2}}{four}+h^{2}}\)
TSA of pyramid = a² + 2a\(\sqrt{\dfrac{a^{two}}{four}+h^{2}}\)
where,

a = Length of the base (foursquare)
l = Camber height
h = Elevation of the pyramid

What Is the Base Area of a Square Pyramid?

The base of a square pyramid is square-shaped. Thus, the base area of square pyramid tin can exist calculated using the formula, Base Area of Pyramid = a², where, a is the length of the base of operations of square pyramid.

How Many Bases Does a Square Pyramid Have?

A square pyramid is a pyramid with a square-shaped base. A foursquare pyramid thus has only ane base of operations.

Which Two Shapes Make up a Foursquare Pyramid?

The base of a square pyramid is a square and its side faces are triangles. Then the ii shapes that make up a square pyramid are square and triangle.

Source: https://www.cuemath.com/measurement/surface-area-of-square-pyramid/

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