Calculating the Total Surface Area of a Prism
Calculating the Total Surface Area of a Prism is a Grade 7 math skill in Illustrative Mathematics, Chapter 7: Angles, Triangles, and Prisms. Students find the total surface area by identifying and calculating the area of each face of the prism and summing them.
Key Concepts
Property The total surface area ($SA$) of a right prism is the sum of its lateral area ($LA$) and the area of its two bases ($2B$). The formula is: $$SA = Ph + 2B$$ where $P$ is the perimeter of the base, $h$ is the height of the prism, and $B$ is the area of the base.
Examples A rectangular prism has a base with dimensions 5 cm by 3 cm and a height of 8 cm. The base perimeter $P = 2(5+3) = 16$ cm and the base area $B = 5 \times 3 = 15$ cm$^2$. The total surface area is $SA = (16)(8) + 2(15) = 128 + 30 = 158$ cm$^2$. A right triangular prism has a height of 12 inches. Its base is a right triangle with legs of 6 inches and 8 inches. The hypotenuse is 10 inches. The base perimeter $P = 6+8+10 = 24$ in and the base area $B = \frac{1}{2}(6)(8) = 24$ in$^2$. The total surface area is $SA = (24)(12) + 2(24) = 288 + 48 = 336$ in$^2$.
Explanation The total surface area represents the entire area of the outside of a 3D object. To calculate it for a prism, you find the area of all the side faces (the lateral area) and add the area of the top and bottom bases. The formula $SA = Ph + 2B$ provides a direct method for this calculation. This skill combines finding the lateral area and the base area into one comprehensive formula for the total surface area.
Common Questions
How do you calculate the total surface area of a prism?
Find the area of each face separately, then add all face areas together. For a rectangular prism with dimensions l, w, and h: SA equals 2(lw plus lh plus wh).
What is a net of a prism and how does it help?
A net is the unfolded flat layout of all faces of a prism. Drawing the net helps you see and calculate each face individually before summing for total surface area.
How many faces does a triangular prism have?
A triangular prism has 5 faces: 2 triangular bases and 3 rectangular lateral faces.
What is an example of calculating prism surface area?
For a rectangular prism 4 cm by 3 cm by 5 cm: SA equals 2(4 times 3 plus 4 times 5 plus 3 times 5) equals 2(12 plus 20 plus 15) equals 2(47) equals 94 square cm.
What chapter covers prism surface area in Illustrative Mathematics Grade 7?
Calculating total surface area of a prism is covered in Chapter 7: Angles, Triangles, and Prisms in Illustrative Mathematics Grade 7.