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This image shows a square pyramid.

What is the surface area of this square pyramid?

A. 36ft^2

B. 72ft^2

C. 78ft^2

D. 84ft^2

arota21  May 23, 2017
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2+0 Answers

 #1
avatar+17601 
+3

The area of the base is 6 x 6 = 36.

 

Since the left-hand triangle of the front side is a 45-45-90 triangle, the length of the dotted line segment (the height) of the front triangle is equal to the distance from the left-hand corner dot to the middle dot of the front side.

Because the length of the base of the front side is 6, the distance from the left-hand corner dot to the middle dot is 3; therefore, the length of the height of the front side is also 3.

 

The area of the front side is 1/2 x base x height = 1/2 x 6 x 3  =  9.

 

The total area = (area of the base) + 4 x (area of the front side)  =  (36) + 4 x (9)  =  72 square feet.

geno3141  May 23, 2017
 #2
avatar+6810 
+1

\(\text{Slant height}=\dfrac{\frac{6}{2}}{\sin45^{\circ}}=3\sqrt2\\ \text{Surface area} = 6^2 + 4\times \dfrac{1}{2}\times 6\times 3\sqrt2\times \sin45^{\circ}\\ \qquad\qquad\;\;\;\;\;\!\;\;\!\!=36 + 36 \\\qquad\qquad\;\;\;\;\;\!\;\;\!\!=72\)

Area of any triangle: a and b are 2 side lengths of the triangle and C is the included angle of a and b, then area = \(\dfrac{ab\sin C}{2}\).

MaxWong  May 24, 2017

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