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Answer:

  • decagon: 238.1 square units
  • prism: 1058 square centimeters

Explanation:

You want the area of a decagon with radius 9 units, and a rectangular prism whose net is shown.

Decagon

The regular decagon can be considered to be comprised of 10 congruent isosceles triangles with side length 9 and vertext angle 360°/10 = 36°.

The area of each triangle is ...

A = 1/2r²sin(θ) . . . . . where r is the side length and θ is the vertex angle

The area of 10 such triangles will be ...

decagon area = 10(1/2)(9²)sin(36°) ≈ 238.1 . . . . square units

The area of the decagon is about 238.1 square units.

Prism

The surface area of the prism is the sum of the areas of the rectangles making up the net. The area of each rectangle is the product of its length and width.

The set of rectangles vertically down the center of the figure has height 2(11 +14) and width 15, so its area is 15·2·(11 +14) = 750 cm².

The two "wings" on either side of that central set of rectangles each have area 11·14 = 154 cm².

The total area of the net is ...

750 cm² + 2(154 cm²) = 1058 cm²

The surface area of the prism is 1058 cm².

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