How to find sa of a pyramid?

We cannot talk about pyramids without mentioning the most famous pyramids in the world, located in Egypt, and built as a tomb for Egyptian kings. Although pyramids are not quite common, their shape is so striking that it seems it is used to make a statement, perhaps that is the reason the Egyptian kings used it as their tomb. The Louvre Museum in Paris, which is the largest art museum in the world, is also shaped like a pyramid. As you can see from the images of the Egyptian Pyramids, a pyramid is a 3-dimensional figure with triangular sides that meet at the edges and at the top to form an apex and it has a polygon as its base.

The polygonal base determines the type of pyramid. Here you see the pyramid with a triangle as its base is a triangular pyramid, the pyramid with a rectangle as its base is a rectangular pyramid, and the pyramid with a pentagon as its base is a pentagonal pyramid. We also have hexagonal pyramids and heptagonal pyramids, and so on.

Let’s take a moment and recall what the volume and surface area of a 3-dimensional figure means.

The volume of a 3-dimensional figure is the measure of how much it can hold, and it is measured in cubic units.

The surface area of a 3-dimensional figure is the measure of the total area that the surface of the figure covers and is measured in square units.

Before we can calculate the volume and surface area of a pyramid, we must know the difference between the height and slant height. The height of a pyramid is the perpendicular length from the apex to the base, and the slant height is the length from the apex to the midpoint of the bottom edge of one of the triangular faces.

Here are the formulas for the volume and surface area of any pyramid.

To calculate the volume and surface area of any pyramid we need B, which represents the area of the base, and p, which represents the perimeter of the base. It is important to note, since the base of a pyramid can be any polygon, we will be using our prior knowledge of finding the area and perimeter of different polygons to calculate the volume and surface area of a pyramid.

Let’s look at an example.

A triangular pyramid has an equilateral triangle as its base with side lengths 6 in and a height of 8 in. What is the volume and surface area of the pyramid?

To find the surface area of a pyramid, we use the formula \(SA=B+\frac{1}{2}ps\), where \(B\) is the area of the base, \(p\) is the perimeter of the base, and \(s\) is the slant height. Since the base is a triangle, we will use the formula for the area of a triangle to find \(B\).

\(\text{Area of a triangle}=\frac{1}{2}bh\), where the \(b\) and \(h\) are the base and height of the triangular base. We will draw a perpendicular line to the base, which is the height of the triangular base and it divides the base of the triangle into 2 equal parts. Since the triangle is now turned into 2 right triangles, we can use the Pythagorean theorem to find the height.

So the Pythagorean theorem is:

And then we’re gonna look at our triangle, which says we have:

We’ll subtract 9 from both sides. This gives us:

And then we’ll square root both sides. Which gives us:

Or:

Therefore, the area of the base can be found by doing:

The perimeter of the base is equal to all the sides added together, so:

Now to solve for surface area, all we have to do is plug these values in for our variables. So our surface area is equal to:

And all this added together is approximately equal to 78.6 square inches.

To find the volume of the triangular pyramid, we need the area of the base, \(B\), and the height of the pyramid, which is 8 inches. So let’s plug this in. So:

Now all we have to do is calculate this out, which is equal to approximately 41.6 cubic inches.

Let’s look at another example. Here is a pyramid with a square base, with side lengths of 5 centimeters. The height of the pyramid is 11 centimeters. What is the volume of the pyramid?

So our volume formula is:

Before we can calculate the volume of the pyramid, we need to find the area of the base. Since the base is a square, we use the formula for the area of a square, which is \(s^{2}\). So to find our area, we’re gonna use \(s^{2}\) and our side length is 5.

Now we can plug this value into our formula.

Then if we plug this into a calculator, we’ll get that it’s approximately equal to 91.67 cubic centimeters.

But when would we ever need this in real life? Well, I’m glad you asked! Take a look at this next example and try it on your own.

The roof of a wooden cottage is shaped like a square-based pyramid. The length of the sides of the square base are 22 feet and the height of the triangular face is 14 feet. Peter wants to paint the roof of his wooden cottage and needs to determine how much paint he needs to buy. We will assume 1 pint of paint covers 400 square feet. How much paint does Peter need to buy?

The roof of the cottage does not include the base of the pyramid. Therefore, we only need to find the area of the 4 triangular faces. This is called the lateral area. So the lateral area is equal to the surface area minus the area of the base. So all we need is:

The perimeter of our square is equal to 4 times the side length.

Which we can plug into a calculator to get:

So we will need 616 square feet to be covered.

If 1 pint of paint covers 400 square feet, we need to divide 616 by 400 to figure out how much paint we need. So:

So, Peter will need to buy 2 pints of paint to cover the roof of the cottage because 1 pint won’t be enough and you can’t get 0.54 of a pint. So 2 pints of paint will cover the whole roof.

The following diagrams show how to find the surface area of a pyramid. Scroll down the page for more examples and solutions.

A pyramid is a solid with a polygonal base and several triangular lateral faces. The lateral faces meet at a common vertex. The number of lateral faces depends on the number of sides of the base. The height of the pyramid is the perpendicular distance from the base to the vertex.

A regular pyramid has a base that is a regular polygon and a vertex that is above the center of the polygon. A pyramid is named after the shape of its base. A rectangular pyramid has a rectangle base. A triangular pyramid has a triangle base.

We can find the surface area of any pyramid by adding up the areas of its lateral faces and its base.

Surface area of any pyramid = area of base + area of each of the lateral faces

If the pyramid is a regular pyramid, we can use the formula for the surface area of a regular pyramid. Surface area of regular pyramid = area of base + 1/2 ps where p is the perimeter of the base and s is the slant height.

If the pyramid is a square pyramid, we can use the formula for the surface area of a square pyramid.

Surface area of square pyramid = b2 + 2bs where b is the length of the base and s is the slant height.

Worksheets: Calculate the volume of square pyramids Calculate the volume of prisms & pyramids

Example: Calculate the surface area of the following pyramid.

Solution: Sketch a net of the above pyramid to visualize the surfaces.

Since the given pyramid is a square pyramid, we can use any of the above formulas.

Using the formula for the surface area of any pyramid:

Area of base = 6 × 6 = 36 cm2

Area of the four triangles = 1/2 × 6 × 12 × 4 = 144 cm2

Total surface area = 36 + 144 = 180 cm2

Using the formula for a regular pyramid

Surface area of regular pyramid = area of base + 1/2 ps

= 6 × 6 + 1/2 × 6 × 4 × 12 = 180 cm2

Using the formula for a square pyramid

Surface area of square pyramid = b2 + 2bs

= 6 × 6 + 2 × 6 × 12 = 180 cm2

How to find the surface area of a pyramid by adding up the area of each surface? Calculate the surface area of the square based pyramid.

Example: Find the surface area of a square pyramid with s = 40in, h = 39in and n = 44in

How to find the surface area of a square pyramid using the formula? Surface area = 2bs + b2 where b is the length of the base and s is the slant height.

Example: What is the surface area of a square pyramid with a base area of 255 square inches and a height of 7 inches?

The Great Pyramid of Khufu, the largest of the pyramid in Giza, was built approximately 4,500 years ago. Today, the height of the pyramid is about 455 feet, which is about 30 feet shorter than it was originally. If you were to walk completely around the base of the pyramid, you would have gone about 3,024 feet. What is the lateral surface area of the great pyramid today?

These videos show how to calculate the surface area of a regular pyramid using the formula: surface area = area of base + 1/2 × perimeter of base × slant height. S = B + 1/2 p l

This video provides a specific example of how to find the surface area of a pyramid, given base edge and height. The base is a pentagon. It shows how to find the apothem and slant height.

How to calculate the surface area of a square pyramid when the slant height is not given?

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