Question

Here is a net made of right triangles and rectangles. All measurements are given in centimeters. Three rectangles in a row measuring 6 by 4, 6 by 5, and 6 by 3. The left rectangle is bordered on the top by a right triangle with side lengths 4, 3, and 5. The right rectangle is bordered on the bottom by a right triangle with side lengths 3, 4, and 5. What is the surface area of the polyhedron? Explain your reasoning.

Answer 1

a. 84 cm² b. The area of the polyhedron is found by finding the sum of the areas of each individual shape.

Explanation:

a. The area of the polyhedron is gotten by finding the area of each individual shape.

The area of the left rectangle with sides 6 by 4 is 6 cm × 4 cm = 24 cm²

The area of the middle rectangle with sides 6 by 5 is 6 cm × 5 cm = 30 cm²

The area of the right rectangle with sides 6 by 3 is 6 cm × 3 cm = 18 cm²

The area of the triangle on the left rectangle is found using heron's formula where area A = √[s(s -a)(s -b)(s -c) where s = (a + b + c)/2 and a, b and c are the sides of the rectangle.

Since the rectangle has sides 4, 3 and 5, a = 4, b = 3 and c = 5

So, s = (a + b + c)/2 = s = (4 + 3 + 5)/2 = 12/2 = 6

A = √[s(s -a)(s -b)(s -c)

= √[6(6 - 4)(6 - 3)(6 - 5)

= √[6(2)(3)(1)

= √36

= 6 cm²

The area of the triangle on the right rectangle is found using heron's formula where area A = √[s(s -a)(s -b)(s -c) where s = (a + b + c)/2 and a, b and c are the sides of the rectangle.

Since the rectangle has sides 3, 4 and 5, a = 3, b = 4 and c = 5

So, s = (a + b + c)/2 = s = (3 + 4 + 5)/2 = 12/2 = 6

A = √[s(s -a)(s -b)(s -c)]

= √[6(6 - 3)(6 - 4)(6 - 5)]

= √[6(3)(2)(1)

= √36

= 6 cm²

The surface area of the polyhedron is the sum of all the areas. So, A = 24 cm² + 30 cm² + 18 cm² + 6 cm² + 6 cm² = 84 cm²

b. Explain your reasoning

The area of the polyhedron is found by finding the sum of the areas of each individual shape.