Calculating the Total Surface Area of a Right Pyramid
Understanding how to calculate the total surface area of a right pyramid is essential in both theoretical and practical applications. This article will guide you through the process, providing a step-by-step breakdown of each calculation.
Introduction
A right pyramid has a polygonal base and its apex is directly above one of the vertices of the base. In this article, we focus on a right pyramid with a square base. The goal is to find the total surface area, considering the area of the base and the four triangular faces.
Calculating the Area of the Base
The first step is to calculate the area of the base, which is a square in this case. Let's denote the side length of the square base as s 10 cm.
Step 1: Calculate the area of the base.
The area of the base can be found using the formula for the area of a square:
Area of the base s^2 10 cm × 10 cm 100 cm^2
Calculating the Area of the Triangular Faces
The next step is to determine the area of the four triangular faces that form the lateral surface of the pyramid. Each triangular face has a base equal to the side of the square base and a height that can be calculated using the Pythagorean theorem.
Finding the Slant Height
The slant height is the height of each triangular face, extending from the midpoint of a base edge to the apex. It can be calculated using the heights provided and the Pythagorean theorem.
Stepwise Calculation
Step 2a: Find the slant height.
The slant height ((l)) can be found using the formula:
[ l sqrt{left(frac{s}{2}right)^2 h^2} ]
Where s/2 5 cm and h 12 cm.
[ l sqrt{5^2 12^2} sqrt{25 144} sqrt{169} 13 text{ cm} ]
Step 2b: Calculate the area of one triangular face.
The area of a triangular face can be calculated using the formula:
[ A frac{1}{2} times text{base} times text{height} ]
Substituting the values we have:
[ A frac{1}{2} times 10 text{ cm} times 13 text{ cm} 65 text{ cm}^2 ]
Step 3: Calculate the total area of the four triangular faces.
Since there are four triangular faces, we multiply the area of one face by 4:
[ text{Total area of triangular faces} 4 times 65 text{ cm}^2 260 text{ cm}^2 ]
Step 4: Calculate the total surface area.
The total surface area ((S)) of the pyramid is the sum of the area of the base and the area of the triangular faces:
[ S text{Area of the base} text{Total area of triangular faces} ]
Substituting the values we have:
[ S 100 text{ cm}^2 260 text{ cm}^2 360 text{ cm}^2 ]
Conclusion
The total surface area of the pyramid is 360 cm^2. This detailed breakdown demonstrates the step-by-step approach to solving such geometry problems involving right pyramids with square bases.
Related Keywords: right pyramid, total surface area, square base