# Understanding the formula for the surface area of ​​a pyramid and examples of problems

Who is confused about finding the formula for the surface area of ​​a pyramid to complete homework from school? You need to understand that solving math problems using the pyramid surface area formula is very easy.

Limas itself has several types of pedestals with related names. For example, a rectangular pyramid at the bottom has four corners. Then the triangular pyramid is the pyramid that has a triangular base.

Come on, see the full explanation of the pyramid surface area formula and examples of problems.

## Build the Limas Room

When talking about this one building, so that you can remember it all the time, try to imagine a historical building called the pyramids in Egypt. The shape of the pyramid building is basically a rectangular pyramid, you can see why it is rectangular.

Apart from the Pyramids of Giza, the shapes of these structures that we usually find in everyday life are rubrics, tents, telephones, towers, and also the top of the temple. Limas itself is a spatial or three-dimensional figure that has a base in the form of polygons or polygons, namely triangles, quadrilaterals, and pentagons.

The sides are triangular and have peaks. The name of this shape is determined based on the shape of the base.

Each spatial shape certainly has its own characteristics to distinguish it from other spatial forms, as well as pyramids. The following are the characteristics of a pyramid shape, namely:

• The top of the pyramid has a sharp point
• The lower part of the pyramid shape is flat
• The perpendicular sides of the pyramid are triangular

## Limas Elements

In addition to the characteristics mentioned above, pyramids also have several elements that must be understood, including:

• The edge is the line of intersection between the 2 sides of the pyramid.
• The corner point where two or more ribs meet.
• The side plane is a plane consisting of a base plane and a vertical side plane.
• The peak point is a meeting point between the blanket and the pyramid blanket.
• The height of the pyramid is the distance between the base and the vertex.

## Various Shapes of Limas

Limas has several geometric shapes based on the shape of the reason, namely:

### 1. Triangular pyramid

A triangular pyramid is a type of pyramid whose base is in the form of a triangle, be it an equilateral triangle, isosceles, or any triangle. The elements of a triangular pyramid are:

• 4 corner points
• 4 side planes
• 6 ribs

A rectangular pyramid is a type of pyramid whose base is rectangular, namely square, rectangle, rhombus, kite, trapezoid, parallelogram, and other rectangular shapes. The following are the elements in the rectangular pyramid, including:

• 5 corner points
• 5 side planes
• 8 ribs

### 3. The pentagonal pyramid

A pentagonal pyramid is a type of pyramid that has the shape of a pentagonal flat base, be it a regular pentagon or an arbitrary pentagon. The following are some of the elements in a pentagonal pyramid:

• 6 corner points
• 6 side planes
• 10 ribs

### 4. Hexagonal pyramid

A hexagonal pyramid is a type of pyramid that has a hexagonal base shape, be it a regular hexagon or an arbitrary hexagon. The following are some of the elements in the hexagonal pyramid:

• 7 corner points
• 7 side planes
• 12 ribs

## The formula for the surface area of ​​a pyramid

Surface area is the total area of ​​a flat shape in the form of a geometric shape. The flat shape that forms a pyramid consists of a base and a triangular upright. So, in general the formula for the surface area of ​​a pyramid is as follows:

The formula for the surface area of ​​a pyramid = area of ​​the base + area of ​​all vertical sides

## Example of the Surface Area of ​​a Limas Problem (Part 1)

In order to better understand the concept of the surface area of ​​a pyramid, here is an example of a problem about the surface area of ​​a pyramid.

### Example Question 1

A rectangular pyramid with a side length of 10 cm and a height of 12 cm, then what is the surface area of ​​the rectangular pyramid?

Is known:

area of ​​base = 10×10 = 100 cm2

pyramid height = 12 cm

The surface area of ​​the pyramid

Completion:

area of ​​base = side x side = 10 x 10 = 100 cm2

The sum of the areas of the perpendiculars = the sum of the areas of the triangles with the perpendicular sides = 4 x the area of ​​the triangle QRT

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area of ​​triangle QRT = 1/2 x QR x BT = 1/2 x 10 x 13 = 65 cm2

the sum of the area of ​​the uprights = 4 x the area of ​​the triangle QRT = 4 x 65 = 260

So, the surface area of ​​the pyramid = 100 + 260 = 360 cm2

### Example Problem 2

The area of ​​the base of the rectangular pyramid is 16 cm2 and the height of the right triangle is 3 cm. Determine the surface area of ​​the triangular pyramid.

Is known:

The area of ​​the base of the pyramid = 16 cm2

height of right triangle = 3 cm

The surface area of ​​the pyramid

Completion:

The surface area of ​​the pyramid = area of ​​the base + sum of the area of ​​the perpendiculars

area of ​​base = 16 cm2

sum of area of ​​perpendiculars = 4 x area of ​​triangle = 4 x (1/2 x 4×3)= 24 cm2

So the surface area of ​​the pyramid = 16 + 24 = 40 cm2

### Example Problem 3

A regular hexagon pyramid has a base area of ​​120 cm2 and a right triangle area of ​​30 cm2. Determine the surface area of ​​the hexagonal pyramid.

Is known:

area of ​​base = 120 cm2

area of ​​right triangle = 30 cm2

The surface area of ​​the pyramid

Resolution:

Surface area = area of ​​the base + sum of the area of ​​the perpendiculars

area of ​​base = 120 cm2

the sum of the area of ​​the uprights = 6 x the area of ​​the right triangle = 6 x 30 cm2 = 180 cm2

So, the surface area of ​​the hexagonal pyramid = 120 + 180 = 300 cm2

## Limas Volume Formula

Limas is also included in the geometric type so that it has a volume. The following is the general formula for the volume of a pyramid.

The volume of the pyramid = 1/3 x base area x height

## Example of a Limas Volume Problem

In order to better understand the use of the pyramid volume formula, here are some examples of questions to find the volume of a pyramid.

### Example Question 1

Find the volume of a triangular pyramid with a base area of ​​50 cm2 and a height of 12 cm.

Is known:

area of ​​base = 50 cm2

pyramid height = 12 cm

Lime volume

Completion:

The volume of the pyramid = 1/3 x the area of ​​the base xt of the pyramid = 1/3 x 50 x 12 = 200 cm3

So, the volume of the rectangular pyramid is 200 cm3

### Example Problem 2

A rectangular pyramid with a side length of 8 cm and a height of 6 cm, what is the volume of the pyramid?

Is known:

side of the quadrilateral = 8 cm

pyramid height = 6 cm

Lime volume

Completion:

The volume of the pyramid = 1/3 x the area of ​​the base xt of the pyramid = 1/3 x ( 8 x 8 ) x 6 = 128 cm3

So, the volume of the rectangular pyramid is 128 cm3 .

Example Problem 3.

A pentagonal pyramid has a known base area of ​​50 cm2 and a height of 15 cm, then what is the volume of the pentagonal pyramid?

Is known:

area of ​​base = 50 cm2

height = 15 cm

The volume of the pentagonal pyramid

Completion:

Volume = 1/3 x base area x height

= 1/3 x 50 x 15

= 250 cm3

So, the volume of the pentagonal pyramid is 250 cm3

## The formula for the surface area of ​​a triangular pyramid

A triangular pyramid has a triangular base. The shape of the base itself can be equilateral, isosceles triangle, arbitrary triangle, or other triangular shape. The triangular pyramid formula is used to find the surface area and volume.

As we have discussed above, when viewed from its shape, this triangular pyramid has the following characteristics:

• Has 4 corner points
• Has 4 sides
• Has 6 ribs.

The formula for the surface area of ​​a triangular pyramid consists of two forms, the first is used to find the surface area and the second is used to find the volume. The formula for the surface area of ​​a pyramid can be symbolized by: “base area + area of ​​the vertical side”.

Meanwhile, the formula for the volume of a pyramid is “1/3 x area x height”.

### Volume Formula:

Volume of triangular pyramid = 1/3 x area of ​​base x height of pyramid.

### Surface Area Formula:

Area of ​​triangular pyramid = area of ​​base + area of ​​casing.

It could also be the area of ​​a triangular pyramid = the sum of the area of ​​the sides.

## The formula for the surface area of ​​a rectangular pyramid

The base and blanket are the most important elements in the formula for the area of ​​a rectangular pyramid. Some of the forming components of a geometric shape, among others, are in the vertical plane. This rectangular pyramid is usually called a rectangular pyramid.

The forming components that are the dominant characteristic of this rectangular pyramid are on its five sides. The five sides are composed of the base side in a quadrilateral shape.

Then, for four upright sides with a triangular shape that becomes the corner. Not only the corners, there are also five corners, namely four at the base and one at the top.

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The number of edges of this rectangular pyramid is 8, with four ribs on the base and the other four on the upright ribs. The formula for the area of ​​a rectangular pyramid is indicated by the area of ​​the base plus the total area of ​​the uprights that exist and are related to the building. You can note the formula below:

V: ⅓ x (sxs) xt

### Surface Area Formula

LP: (sxs) + (4 x area of ​​the perpendicular)

There is also an easy way to determine the validity of the area formula for a rectangular pyramid, namely by determining the midpoint of the point of intersection which is usually marked with the letter o. Even though this method is actually more effective for cubes, which are a combination of six pyramids at once.

## The formula for the surface area of ​​a hexagonal pyramid

This hexagon pyramid is a three-dimensional geometric figure bounded by a hexagon-shaped base and six triangular upright sides. At a glance, maybe some people will find it difficult to distinguish it from a hexagonal prism and a hexagonal pyramid. However, both have different forms.

Judging from its shape, a hexagonal prism is a three-dimensional shape bounded by a base and lid that are identical in shape to a hexagon and also to have vertical sides in a rectangular shape. That is why hexagonal pyramids have the same properties as other pyramids. The difference is only in the corner points, the following are the general properties of a hexagonal pyramid:

• It has a triangular straight side
• The side of the base is in the shape of a polygon
• Has one breaking point
• The naming of the pyramid depends on the shape of the base

Before completing the example questions related to the hexagonal pyramid. It would be better to know in advance about the formulas for pyramids, including the surface and volume formulas for pyramids.

### Surface Area Formula

Surface area = top area + sheath area of ​​the pyramid

= area of ​​base + (nx area of ​​triangle)

### Volume Formulas

Volume = 1/3 x base area x height

= 1/3 x W x h

## Examples of Problems Calculating the Surface Area of ​​a Pyramid (Part 2)

To make it easier to understand, here is an example of a collection of semicircle formula questions.

Complete with the formula for the circumference of a semicircle, to the area of ​​a flat circle.

### 1. Example Question 1

A T.ABCD pyramid. If the volume of the pyramid is 2.304 cm3 and the height of the pyramid is 27 cm. What is the surface area of ​​the pyramid?

Discussion:

It is known that the volume of pyramid T.ABCD = 2.304 cm3 and the height of pyramid T.ABCD = 27 cm

Wanted: Surface area of ​​the pyramid T.ABCD?

• Step 1. Find the T.ABCD Volume

The volume of the pyramid = area of ​​the high base 2.304 = 1/3 x area of ​​the base xt

2.304 = 1/3 x area of ​​base xt

2.304 = 1/3 x base area x 27

Base area = 2.304 / 9

The area of ​​the base of the pyramid = 256

Thus, side of the square AB = square root of 256 = 16 cm

• Step 2. Find the surface area of ​​the pyramid

The surface area of ​​the pyramid = area of ​​the base + area of ​​the perpendicular

Lp = (sxs) + (4 x ½ xsx t. blanket)

Lp = (16 x 16) + ( 2 x 16 x 27)

Lp = 256 + 864 = 1.120 square cm

So the surface area of ​​the pyramid T. ABCD is 1.120 square cm.

### 2. Sample Question 2

Determine the surface area and volume of a rectangular pyramid with a square base that has a side of 14 cm and a height of 6 cm, and the height of the right triangle is 8 cm!

Discussion:

Given: The base side is 14 cm, the height of the pyramid is 6 cm, and the height of the right triangle is 8 cm.

Wanted: Surface area and volume of a rectangular pyramid

• Step 1. Find the surface area of ​​the pyramid

The surface area of ​​the pyramid

= Area of ​​the base + 4 x area of ​​the perpendicular

= (14 cm x 14 cm) + (1/2 x 14 cm x 8 cm)

= 196 cm2 + 56 cm2

= 252 cm2

• Step 2. Find the volume of the pyramid

Lime volume

= 1/3 x base area x height

= 1/3 x 196 cm2 x 6 cm

= 392 cm2

So, the surface area of ​​the pyramid is 252 cm3 and the volume is 392 cm3.

### 3. Sample Question 3

The length of the base of a regular hexagonal pyramid is 8 cm.

If the height of a regular triangle is 8 cm and the height of the triangle on the right side is 15 cm. So, what is the surface area of ​​the pyramid?

The area of ​​the base of the pyramid is determined from 6 congruent equilateral triangles, namely:

t = √8² – 4²

t =√64 – 16

t = √48

t = 6.9 cm

After getting the height of the hexagonal pyramid, here’s how to calculate the area of ​​the base of the pyramid, namely:

L = 6 x ( 1/2 x base x height)

L = 6 x (1/2 x 8 x 6.9)

L = 6 x 27.6

L = 165.6 cm²

So, the surface area of ​​the hexagonal pyramid is 165.6 cm².

Thus the discussion of the surface area of ​​the pyramid and also the formula for the volume of the pyramid. Hopefully all the discussion above is useful and also adds to Sinaumed’s’ insight.

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