Calculating the Total Surface Area of a Square-Based Pyramid: A Comprehensive Guide
Understanding the total surface area (TSA) of a square-based pyramid is crucial for various applications in geometry, engineering, and design. This article will walk you through the process of calculating the TSA of a square-based pyramid given its slant height and vertical height.
What is a Square-Based Pyramid?
A square-based pyramid is a three-dimensional geometric shape with a square base and four triangular faces that meet at a common vertex (apex). The slant height is the height of these triangular faces, while the vertical height is the distance from the center of the base to the apex.
Given Information
Slant height (l) 10 cm Vertical height (h) 8 cmStep-by-Step Calculation
Find the Length of the Base: To find the length of one side of the base (a), we can use the Pythagorean theorem. The relationship between the vertical height, half the base, and the slant height is given by:l2 h2 (a/2)2
Substitute the Known Values:102 82 (a/2)2
100 64 (a/2)2
(a/2)2 100 - 64
(a/2)2 36
a/2 6 rarr; a 12 cm
Calculate the Base Area
The area of the base (Abase) is given by:
Abase a2 122 144 cm2
Calculate the Area of the Triangular Faces
Each triangular face has a base of a and a height of the slant height (l). The area of one triangular face (Atriangle) is given by:
Atriangle 1/2 times; a times; l 1/2 times; 12 times; 10 60 cm2
Since there are four triangular faces, the total area of the triangular faces (Atriangles) is:
Atriangles 4 times; Atriangle 4 times; 60 240 cm2
Calculate the Total Surface Area
The total surface area (Atotal) of the pyramid is the sum of the base area and the lateral area:
Atotal Abase Atriangles 144 240 384 cm2
Conclusion
The total surface area of the square-based pyramid with a slant height of 10 cm and a vertical height of 8 cm is 384 cm2.