Question

station 9: square pyramid. find the surface area of the figures below. show all steps in an organized way. figures are not drawn to scale.

Explanation:

Step1: Recall surface - area formula for square pyramid

The surface area formula for a square pyramid is $SA = B+4\times(\frac{1}{2}bh)$, where $B$ is the area of the base, $b$ is the base of a triangular face, and $h$ is the slant - height of the triangular face.

Step2: Calculate base area

For a square base with side length $s$, $B=s^{2}$.

Step3: Calculate area of one triangular face

The area of one triangular face is $A_{triangle}=\frac{1}{2}bh$, where $b$ is the side length of the base and $h$ is the slant - height.

Step4: Calculate total surface area

Add the base area and the areas of the four triangular faces.

Let's take the first square pyramid (top - left) with base side length $s = 6$ ft and slant - height $h=7$ ft:

• Calculate base area: $B = 6^{2}=36$ square feet.
• Calculate area of one triangular face: $A_{triangle}=\frac{1}{2}\times6\times7 = 21$ square feet.
• Calculate total surface area: $SA=36 + 4\times21=36 + 84 = 120$ square feet.

For the second square pyramid (top - right) with base side length $s = 5$ mm and slant - height $h = 5.7$ mm:

• Base area $B = 5^{2}=25$ square mm.
• Area of one triangular face $A_{triangle}=\frac{1}{2}\times5\times5.7 = 14.25$ square mm.
• Total surface area $SA=25+4\times14.25=25 + 57=82$ square mm.

For the third square pyramid (middle - left) with base side length $s = 5$ yd and slant - height $h = 7.6$ yd:

• Base area $B = 5^{2}=25$ square yd.
• Area of one triangular face $A_{triangle}=\frac{1}{2}\times5\times7.6 = 19$ square yd.
• Total surface area $SA=25+4\times19=25 + 76 = 101$ square yd.

For the fourth square pyramid (middle - right) with base side length $s = 4$ yd and slant - height $h = 5.3$ yd:

• Base area $B = 4^{2}=16$ square yd.
• Area of one triangular face $A_{triangle}=\frac{1}{2}\times4\times5.3 = 10.6$ square yd.
• Total surface area $SA=16+4\times10.6=16 + 42.4 = 58.4$ square yd.

For the fifth square pyramid (bottom - left) with base side length $s = 5$ mi and slant - height $h = 4.7$ mi:

• Base area $B = 5^{2}=25$ square mi.
• Area of one triangular face $A_{triangle}=\frac{1}{2}\times5\times4.7 = 11.75$ square mi.
• Total surface area $SA=25+4\times11.75=25 + 47 = 72$ square mi.

For the sixth square pyramid (bottom - right) with base side length $s = 3$ yd and slant - height $h = 6.2$ yd:

• Base area $B = 3^{2}=9$ square yd.
• Area of one triangular face $A_{triangle}=\frac{1}{2}\times3\times6.2 = 9.3$ square yd.
• Total surface area $SA=9+4\times9.3=9 + 37.2 = 46.2$ square yd.

Answer:

1. 120 square feet
2. 82 square mm
3. 101 square yd
4. 58.4 square yd
5. 72 square mi
6. 46.2 square yd