Question

how can you calculate the surface area of a prism or a pyramid from a picture?

Explanation:

Step1: Identify all faces

For a prism or pyramid from a picture, first identify each individual face. For a rectangular - prism, there are 6 faces; for a triangular - prism, there are 5 faces; for a pyramid, the number of faces depends on the base (a triangular - pyramid has 4 faces, a square - pyramid has 5 faces etc.).

Step2: Determine face - shapes

Determine the shape of each face. Common shapes are rectangles, triangles. For example, in a rectangular - prism, all faces are rectangles; in a triangular - prism, there are triangles and rectangles; in a pyramid, the base can be a polygon (triangle, square etc.) and the lateral faces are triangles.

Step3: Calculate area of each face

Use the appropriate area formulas. For a rectangle with length $l$ and width $w$, the area formula is $A = lw$. For a triangle with base $b$ and height $h$, the area formula is $A=\frac{1}{2}bh$. Calculate the area of each face using the dimensions from the picture.

Step4: Sum up the areas

Add up the areas of all the faces to get the surface area of the prism or pyramid.

Let's take the first rectangular - prism with length $l = 3$, width $w = 2$ and height $h = 4$:
The two faces with dimensions $3\times2$ have area $A_1=3\times2 = 6$ each.
The two faces with dimensions $3\times4$ have area $A_2=3\times4 = 12$ each.
The two faces with dimensions $2\times4$ have area $A_3=2\times4 = 8$ each.
The surface area $S=2\times6 + 2\times12+2\times8=12 + 24+16 = 52$.

For the second prism (a triangular - prism with right - angled triangular bases):
The two triangular bases with base $b = 5$ and height $h = 8$ have area $A_{base}=\frac{1}{2}\times5\times8 = 20$ each.
The three rectangular faces: one with dimensions $5\times13$, area $A_4 = 5\times13=65$; one with dimensions $8\times13$, area $A_5 = 8\times13 = 104$; one with dimensions $12\times13$, area $A_6=12\times13 = 156$.
The surface area $S = 2\times20+65 + 104+156=40+65 + 104+156=365$.

For the pyramid (assuming it's a triangular - pyramid with base as an equilateral triangle of side $a = 3$ and height of lateral face (slant height) $l_s=6$):
The base area $A_{base}=\frac{\sqrt{3}}{4}a^{2}=\frac{\sqrt{3}}{4}\times3^{2}=\frac{9\sqrt{3}}{4}\approx3.9$ (using the formula for the area of an equilateral triangle).
The three lateral - face areas, each with area $A_{lateral}=\frac{1}{2}\times a\times l_s=\frac{1}{2}\times3\times6 = 9$.
The surface area $S=\frac{9\sqrt{3}}{4}+3\times9=\frac{9\sqrt{3}}{4}+27\approx3.9 + 27=30.9$.

Answer:

To calculate the surface area of a prism or pyramid from a picture, identify all faces, determine their shapes, calculate the area of each face using appropriate formulas, and sum up the areas of all faces. For the given rectangular - prism with dimensions $l = 3$, $w = 2$, $h = 4$, the surface area is 52; for the given triangular - prism with relevant dimensions, the surface area is 365; for the given triangular - pyramid with relevant dimensions, the surface area is approximately 30.9.