Triangle In A Pentagon

Geometry Level 2

A regular pentagon with side length 5 is split into a quadrilateral and a triangle when two of the vertices are connected. What is the area of the triangle?

Give your answer to 2 decimal places.


The answer is 11.89.

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Janae Pritchett by Janae Pritchett

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2 solutions

Abc Xyz
Mar 8, 2016

Since it is a regular pentagon,the angles are 108 degrees and the sides are 5.

Therefore the area of the yellow triangle can be found with the 1 2 sin ⁡ θ × a × b \frac{1}{2} \sin \theta\times a \times b

Thus the answer is 1 2 sin ⁡ 108 × 5 × 5 \frac{1}{2}\sin 108\times 5\times 5 which comes to 11.89 \boxed{11.89} (approx).

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Ashutosh Malaviya
Mar 13, 2016

Theta = 108 degrees; Area of yellow triangle = (Height * Base)/2; Height = 5 Cos(Theta/2); Base = (5 Sin(Theta/2) + 5 Sin(Theta/2)); Area = (10 Sin(Theta/2)) (5 Cos(Theta/2))/2 = 25 Sin(54 degrees) Cos(54 degrees) = 11.87

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