BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon102 Learners

Last updated on July 31st, 2025

Math Whiteboard Illustration

Surface Area of Right Pyramid

Professor Greenline Explaining Math Concepts

A right pyramid is a 3-dimensional shape with a polygonal base and triangular faces that meet at a common point, called the apex. The surface area of a right pyramid is the total area covered by its outer surface. This includes both the lateral surface area (the sum of the areas of the triangular faces) and the base area. In this article, we will learn about the surface area of a right pyramid.

Surface Area of Right Pyramid for Saudi Students
Professor Greenline from BrightChamps

What is the Surface Area of a Right Pyramid?

The surface area of a right pyramid is the total area occupied by the boundary or surface of the pyramid. It is measured in square units.

 

A right pyramid is a 3D shape with a polygonal base and triangular faces that meet at a single point called the apex. The apex is directly above the center of the base, making the pyramid symmetrical.

 

The surface area of a right pyramid includes both the lateral surface area (the triangular faces) and the base area.

Professor Greenline from BrightChamps

Surface Area of a Right Pyramid Formula

A right pyramid has a lateral surface, and it has two types of surface areas: the lateral surface area and the total surface area.

 

The lateral surface area is the sum of the areas of the triangular faces, while the total surface area includes both the lateral surface area and the base area.

 

A right pyramid has two types of surface areas: Lateral Surface Area of a Right Pyramid Total Surface Area of a Right Pyramid

Professor Greenline from BrightChamps

Lateral Surface Area of a Right Pyramid

The lateral surface area of a right pyramid is the sum of the areas of the triangular faces that connect the base to the apex. The formula for the lateral surface area depends on the shape of the base.

 

For a pyramid with a regular polygon base, the formula is: Lateral Surface Area = 1/2 × Perimeter × Slant Height

 

Here, the perimeter is the perimeter of the base of the pyramid. The slant height is the height of each triangular face from the base to the apex.

Professor Greenline from BrightChamps

Total Surface Area of a Right Pyramid

The total surface area of a right pyramid is the sum of the lateral surface area and the base area.

 

The formula is: Total Surface Area = Lateral Surface Area + Base Area

 

Derivation of the Total Surface Area of a Right Pyramid

 

To find the total surface area of a right pyramid, calculate the lateral surface area using the perimeter and slant height, then add the area of the base.

 

Let us consider a right pyramid with a regular polygon base.

 

Total surface area of a right pyramid = base area + lateral surface area

 

Lateral surface area = 1/2 × perimeter × slant height

 

Substituting the formulas into the total surface area, Total surface area = base area + 1/2 × perimeter × slant height

Professor Greenline from BrightChamps

Volume of a Right Pyramid

The volume of a right pyramid shows how much space is inside it. It tells us how much space is inside the pyramid or how much it can hold.

 

The volume of a right pyramid can be found by using the formula: Volume = 1/3 × Base Area × Height (cubic unit)

Max Pointing Out Common Math Mistakes

Confusion between Lateral Surface Area and Total Surface Area

Students assume that the lateral surface area and the total surface area of a right pyramid are the same. This confusion arises because both involve the slant height. Always remember that the lateral surface area is used only for the triangular faces, while the total surface area includes both the triangular faces and the base.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Using height instead of slant height

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Some students mistakenly use the vertical height instead of the slant height when finding the lateral surface area. Remember the formula for lateral surface area is 1/2 × perimeter × slant height, so always use the slant height.

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Incorrect calculation of the base perimeter

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students may miscalculate the perimeter of the base, especially if the base is not a simple shape like a square. Ensure you correctly add the lengths of all sides of the base to find the perimeter.

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Forgetting to include the base area in the total surface area

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students often calculate only the lateral surface and forget to add the base area. Always include both parts when calculating the total surface area.

 

Total Surface Area = Lateral Surface Area + Base Area.

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Assuming all triangular faces are the same

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Some students mistakenly believe that all triangular faces in a right pyramid are the same, which may not be true if the base is not regular. Ensure you calculate each triangle's area correctly.

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Solved Examples of Surface Area of Right Pyramid

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Find the lateral surface area of a right pyramid with a square base where each side is 6 cm and the slant height is 10 cm.

arrow-right
Max from BrightChamps Saying "Hey"
Hey!

Lateral Surface Area = 120 cm²

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Given the side of the square base = 6 cm, slant height = 10 cm. Perimeter = 4 × 6 = 24 cm. Use the formula: Lateral Surface Area = 1/2 × Perimeter × Slant Height = 1/2 × 24 × 10 = 12 × 10 = 120 cm²

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

Find the total surface area of a right pyramid with a triangular base with sides 5 cm each and a slant height of 8 cm.

Explanation

Total Surface Area = 65.48 cm²

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 2

Perimeter of the base = 3 × 5 = 15 cm. Lateral Surface Area = 1/2 × 15 × 8 = 60 cm². Base Area (for an equilateral triangle) = (√3/4) × 5² = 10.83 cm². Total Surface Area = Lateral Surface Area + Base Area = 60 + 10.83 = 70.83 cm²

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

A right pyramid has a rectangular base with dimensions 4 cm by 6 cm and a slant height of 9 cm. Find the total surface area.

Explanation

Total Surface Area = 130 cm²

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 3

Perimeter of the base = 2(4 + 6) = 20 cm. Lateral Surface Area = 1/2 × 20 × 9 = 90 cm². Base Area = 4 × 6 = 24 cm². Total Surface Area = Lateral Surface Area + Base Area = 90 + 24 = 114 cm²

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

Find the lateral surface area of a right pyramid with a hexagonal base where each side is 3 cm and the slant height is 7 cm.

Explanation

Lateral Surface Area = 63 cm²

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 4

Perimeter of the hexagonal base = 6 × 3 = 18 cm. Use the formula: Lateral Surface Area = 1/2 × Perimeter × Slant Height = 1/2 × 18 × 7 = 9 × 7 = 63 cm²

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The slant height of a right pyramid is 12 cm, and its lateral surface area is 180 cm². Find the perimeter of the base.

Explanation

Perimeter = 30 cm

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Ray Thinking Deeply About Math Problems

It is the total area that covers the outside of a right pyramid, including its triangular faces and the base.

1.What are the two types of surface area in a right pyramid?

Math FAQ Answers Dropdown Arrow

2.What is the difference between slant height and height?

Math FAQ Answers Dropdown Arrow

3.How do you find the lateral surface area of a right pyramid?

Math FAQ Answers Dropdown Arrow

4.What unit is surface area measured in?

Math FAQ Answers Dropdown Arrow
Professor Greenline from BrightChamps

Common Mistakes and How to Avoid Them in the Surface Area of a Right Pyramid

Students often make mistakes while calculating the surface area of a right pyramid, which leads to wrong answers. Below are some common mistakes and the ways to avoid them.

Math Teacher Background Image
Math Teacher Image

Seyed Ali Fathima S

About the Author

Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.

Max, the Girl Character from BrightChamps

Fun Fact

: She has songs for each table which helps her to remember the tables

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
UAE - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom