Question

find the total surface area of:
a image of a triangular prism with dimensions 3 cm, 4 cm, 5 cm, 6 cm
b image of a triangular prism with dimensions 10 m, 6 m, 12 m

Explanation:

Step1: Calculate the area of the triangular base

For part a:
The base of the triangle \(b = 3\mathrm{cm}\), height \(h=4\mathrm{cm}\).
The area of a triangle \(A_{\triangle}=\frac{1}{2}\times b\times h=\frac{1}{2}\times3\times4 = 6\mathrm{cm}^{2}\)

Step2: Calculate the perimeter of the triangular base

The sides of the triangle are \(3\mathrm{cm}\), \(4\mathrm{cm}\), \(5\mathrm{cm}\).
The perimeter \(P=3 + 4+5=12\mathrm{cm}\)

Step3: Calculate the lateral surface area

The length of the prism \(l = 6\mathrm{cm}\)
The lateral surface area \(A_{lateral}=P\times l=12\times6 = 72\mathrm{cm}^{2}\)

Step4: Calculate the total surface area

The total surface area \(A_{total}=2\times A_{\triangle}+A_{lateral}\)
\(A_{total}=2\times6+72=12 + 72=84\mathrm{cm}^{2}\)

For part b:

Step1: Calculate the area of the triangular base

Using Heron's formula. Let \(a = 10\mathrm{m}\), \(b = 12\mathrm{m}\), \(c = 10\mathrm{m}\)
The semi - perimeter \(s=\frac{a + b + c}{2}=\frac{10+12 + 10}{2}=16\mathrm{m}\)
The area of the triangle \(A_{\triangle}=\sqrt{s(s - a)(s - b)(s - c)}=\sqrt{16(16 - 10)(16 - 12)(16 - 10)}=\sqrt{16\times6\times4\times6}=\sqrt{2304}=48\mathrm{m}^{2}\)

Step2: Calculate the perimeter of the triangular base

The perimeter \(P=10+12 + 10=32\mathrm{m}\)

Step3: Calculate the lateral surface area

The height of the prism \(h = 6\mathrm{m}\)
The lateral surface area \(A_{lateral}=P\times h=32\times6=192\mathrm{m}^{2}\)

Step4: Calculate the total surface area

The total surface area \(A_{total}=2\times A_{\triangle}+A_{lateral}\)
\(A_{total}=2\times48+192=96+192 = 288\mathrm{m}^{2}\)

Answer:

a. \(84\mathrm{cm}^{2}\)
b. \(288\mathrm{m}^{2}\)