how to find the surface area of a pyramid

Surface Surface area of a Foursquare Pyramid

In this department, we will larn about the surface surface area of a square pyramid. A pyramid is a three-D object whose all side faces are congruent triangles and whereas its base tin be any polygon. Ane side of each of these triangles coincides with one side of the base polygon. A square pyramid is a pyramid whose base of operations is a square. The pyramids are named according to the shape of their bases. Just like other iii-dimensional shapes, a foursquare pyramid besides has 2 types of areas.

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

Allow us learn about the surface surface area of a square pyramid forth with the formula and a few solved examples here. You tin can find a few do questions in the end.

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

What is the Surface Expanse of a Square Pyramid?

The word "surface" ways " the exterior or outside part of an object or body". So, the total surface expanse of a square pyramid is the sum of the areas of its lateral faces and its base. We know that a square pyramid has:

a base which is a square.
iv side faces, each of which is a triangle.

All these triangles are isosceles and coinciding, 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 of operations area which is square.

Formula of Surface Surface area of a Square Pyramid

Let us consider a square pyramid whose base of operations's length (square's side length) is 'a' and the height of each side face (triangle) is 'l' (this is too known as the slant acme). i.eastward., the base of operations and tiptop of each of the 4 triangular faces are 'a' and 'fifty' respectively. So the base surface area of the pyramid which is a square is a × a = a² and the area of each such triangular face is i/2 × a × l. And so the sum of areas of all 4 triangular faces is 4 ( ½ al) = 2 al. Let us now empathise the formulas to summate the lateral and full surface area of a square pyramid using summit and slant height.

Total Expanse of Square Pyramid Using Slant Tiptop

The total surface area of a square pyramid is the total surface area covered by the four triangular faces and a square base. The total surface surface area of a square pyramid using slant superlative can be given by the formula,
Surface surface area of a foursquare pyramid = a²+ 2al
where,

a = base length of foursquare pyramid
fifty = slant tiptop or pinnacle of each side confront

Total Surface Area of a Foursquare Pyramid Using Height

Permit us assume that the height of the pyramid (altitude) be 'h'. Then by applying Pythagoras theorem (y'all can refer to the below figure),

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

Substituting this in the above formula,

The surface area of a foursquare pyramid = a²+ 2al = a^two+ 2a\(\sqrt{\dfrac{a^{2}}{4}+h^{ii}}\)

Note: \(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\) can be simplified every bit \(\dfrac i two \sqrt{a^two+4h^2}\). Thus, the formula of surface area of a square pyramid can be written as aⁱⁱ+ 2a \(\left(\dfrac 1 2 \sqrt{a^2+4h^ii}\right)\) = a²+ a\( \sqrt{a^2+4h^ii}\).

Lateral Surface Expanse of a Square Pyramid

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

a = base length of square pyramid
l = camber height or pinnacle of each side face
h = height of square pyramid

How to Calculate Surface Area of Foursquare Pyramid?

The surface area of a square pyramid can be calculated by representing the 3D figure into a second net. After expanding the 3D figure into a 2nd net nosotros will get one square and four triangles.

Surface Area of a Square Pyramid
The following steps are used to calculate the surface surface area of a foursquare pyramid :

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

Now, that we have seen the formula and method to summate the surface surface area of a square pyramid, let the states have a look at a few solved examples to understand it meliorate.

Examples on Surface Area of a Foursquare Pyramid

Example 1: Discover the expanse of a square pyramid of slant acme 15 units and base length 12 units.

Solution

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

Its camber pinnacle is, l = 15 units.

The surface area = a² +2al = 12²+ii (12) (15) = 504 unitsⁱⁱ

Answer: The surface expanse of the given square pyramid is 504 units².
Example 2: The peak of a square pyramid is 25 units and the base area of a foursquare pyramid is 256 square units. Detect its surface area.

Solution

Allow the side of the base of operations (foursquare) be 'a' units.

Then it is given that a² = 256 ⇒ a = 16 units.

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

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

= 16^two+ii (16) \(\sqrt{\dfrac{16^{2}}{four}+25^{2}}\) ≈ 1095.96 foursquare units.

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

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Exercise Questions on Surface Area of a Square Pyramid

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FAQs on Surface Area of a Square 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 four triangular side faces with the base of operations area of the square pyramid. If a, h, and l are the base length, the height of the pyramid, and slant height respectively, and so the surface area of the square pyramid = a^two+ 2al (or) a²+2a \(\sqrt{\dfrac{a^{2}}{4}+h^{2}}\).

How Do You Observe the Lateral Expanse of a Square Pyramid?

To observe the lateral area of a square pyramid, observe the area of one side face (triangle) and multiply information technology past iv. If a and l are the base length and the slant height of a square pyramid, then the lateral area of the square pyramid = iv (½ × a × fifty) = 2al.

If h is the height of the pyramid, then the lateral expanse = 2a \(\sqrt{\dfrac{a^{two}}{iv}+h^{2}}\).

What Is the Surface area of One of the Triangular Faces of a Square Pyramid?

If a and l are the base length and the slant tiptop of a square pyramid, then the area of i of the iv triangular side faces is, ½ × a × 50.

How Do You Notice the Lateral Area and Surface Area of a Foursquare Pyramid?

The lateral surface area of a square pyramid is the sum of the areas of the side faces simply, whereas the surface expanse is the lateral area + expanse of the base. The lateral area of a square pyramid = 2al (or) 2a\(\sqrt{\dfrac{a^{ii}}{4}+h^{2}}\).

To go the full surface expanse, we need to add the expanse of the base (which is a²) to each of these formulas. The full surface expanse of a foursquare pyramid = a² + 2al (or) aⁱⁱ + 2a\(\sqrt{\dfrac{a^{2}}{four}+h^{ii}}\).
where,

a = Length of the base (square)
l = Slant height
h = Height of the pyramid

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

We know, slant acme of a square pyramid is given in terms of height and base length by the formula, \(l = \sqrt{\dfrac{a^{2}}{4}+h^{two}}\). Nosotros tin calculate the slant height from the given height and base of operations length and apply the formula for surface area of square pyramid every bit,
LSA of pyramid = 2a\(\sqrt{\dfrac{a^{2}}{four}+h^{2}}\)
TSA of pyramid = a² + 2a\(\sqrt{\dfrac{a^{ii}}{iv}+h^{2}}\)
where,

a = Length of the base (foursquare)
fifty = Slant tiptop
h = Height of the pyramid

What Is the Base of operations Area of a Foursquare Pyramid?

The base of a square pyramid is foursquare-shaped. Thus, the base surface area of foursquare pyramid can be calculated using the formula, Base Area of Pyramid = a^two, where, a is the length of the base of square pyramid.

How Many Bases Does a Square Pyramid Have?

A foursquare pyramid is a pyramid with a foursquare-shaped base. A square pyramid thus has but one base of operations.

Which Two Shapes Make up a Square Pyramid?

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

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

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