Find the surface area

Geometry Level pending

Shown above is a regular pyramid, the lower base is a 2 × 4 2 \times 4 rectangle and the upper base is a square with side length of 1 1 . If the height is 5 5 , find the surface area of the pyramid correct to three decimal places.


The answer is 49.785.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Consider my diagram.

a = 4 − 1 2 = 3 2 a=\dfrac{4-1}{2}=\dfrac{3}{2}

b = 5 2 + ( 3 2 ) 2 = 1 2 109 b=\sqrt{5^2+\left(\dfrac{3}{2}\right)^2}=\dfrac{1}{2}\sqrt{109}

c = 2 − 1 2 = 1 2 c=\dfrac{2-1}{2}=\dfrac{1}{2}

d = 5 2 + ( 1 2 ) 2 = 1 2 101 d=\sqrt{5^2+\left(\dfrac{1}{2}\right)^2}=\dfrac{1}{2}\sqrt{101}

surface area = area of upper base + area of lower base + area of four trapezoids \text{surface area = area of upper base + area of lower base + area of four trapezoids}

surface area = 1 2 + 2 ( 4 ) + 2 ( 1 2 ) ( 1 + 2 ) ( 1 2 109 ) + 2 ( 1 2 ) ( 1 + 4 ) ( 1 2 101 ) ≈ 49.785 \text{surface area}=1^2 + 2(4)+2\left(\dfrac{1}{2}\right)(1+2)\left(\dfrac{1}{2}\sqrt{109}\right)+2\left(\dfrac{1}{2}\right)(1+4)\left(\dfrac{1}{2}\sqrt{101}\right) \approx \boxed{ 49.785}

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