Question

surface area of solids using nets
using the net of each solid shape, find its surface area.
(first problem with triangular prism net and related areas: area of j =
area of k =
area of l =
area of m =
area of n =
surface area =
21 (second problem with pyramid net and related areas: area of u =
area of v =
area of w =
area of x =
area of y =
surface area =

1. (third problem with rectangular prism net and related areas: area of a =

area of b =
area of c =
area of d =
area of e =
surface area = )

Explanation:

Step1: Analyze the first solid (triangular prism)

The triangular base has legs 6 ft, 8 ft, hypotenuse 10 ft (by Pythagoras: \(6^2 + 8^2 = 10^2\)). The height of the prism is 10 ft.

• Area of \(J\) (triangle): \(\frac{1}{2} \times 6 \times 8 = 24\) sq ft.
• Area of \(K\) (rectangle): \(8 \times 10 = 80\) sq ft.
• Area of \(L\) (rectangle): \(6 \times 10 = 60\) sq ft.
• Area of \(M\) (rectangle): \(10 \times 10 = 100\) sq ft.
• Area of \(N\) (triangle): same as \(J\), \(24\) sq ft.

Step2: Calculate surface area

Sum all areas: \(24 + 80 + 60 + 100 + 24 = 288\) sq ft.

Step1: Analyze the second solid (square pyramid)

Base is square with side 16 yd. Slant height (height of triangular face) is 15 yd.

• Area of \(U, V, W, X, Y\): \(U, V, X, Y\) are triangles, \(W\) is square.
• Area of \(W\) (square): \(16 \times 16 = 256\) sq yd.
• Area of one triangular face (e.g., \(U\)): \(\frac{1}{2} \times 16 \times 15 = 120\) sq yd. There are 4 triangular faces (\(U, V, X, Y\)).

Step2: Calculate surface area

Sum: \(256 + 4 \times 120 = 256 + 480 = 736\) sq yd.

Step1: Analyze the third solid (rectangular prism)

Dimensions: Let's assume length \(l = 10\) in, width \(w = 5\) in, height \(h = 12\) in (from the net).

• Area of each pair of faces:
• \(A, F\): \(5 \times 10 = 50\) each.
• \(B, E\): \(10 \times 12 = 120\) each.
• \(C, D\): \(5 \times 12 = 60\) each.

Step2: Calculate surface area

Sum: \(2(50 + 120 + 60) = 2 \times 230 = 460\) sq in.

Answer:

1. Surface Area of triangular prism: \(\boldsymbol{288}\) sq ft.
2. Surface Area of square pyramid: \(\boldsymbol{736}\) sq yd.
3. Surface Area of rectangular prism: \(\boldsymbol{460}\) sq in. (Note: Dimensions assumed from net; adjust if actual dimensions differ.)