Surface area of composite figure

Question

https://ibb.co/Z6p8Vsnf

Answer 1

If you know how to calculate the surface area of a cylinder, the only tricky thing about this question is to calculate the area of the two (identical) triangles and to subtract it from the area of the cylinder.

Let's first calculate the surface area of the cylinder, which has two components:

1 - The curved surface: 2 x (pi) x r x h

2 - The area of two circles: 2 x (pi) x r x r

where r is the radius of the circle, h is the height of the cylinder and pi = 3.14. r = 12/2 = 6 and h = 45

So we get:

For 1: 2 x (3.14) x 6 x 45 = 1695.6

For 2: 2 x (3.14) x 6 x 6 = 226.08

Area of the cylinder (S) = 1695.6 + 226.08 = 1921.68

Now let's take a look at the triangles that are cut out from the two circular faces of the cylinder.

The triangles are isosceles. Therefore, the height of each triangle divides the base into two equal parts. Also , each half of a triangle is a right triangle, and therefore we can calculate the height using the Pythagoras theorem as follows:

Hypotenuse = 5

Opposite side = 4

Adjacent side (h) = ?

5 x 5 = (4 x 4) + (h x h)

h = sqrt(9) = 3

Base of each triangle is given and is 8 so the area of one of the triangles (A) is then A = 1/2 x 8 x 3 = 12

Since there are 2 identical triangle cut outs we need to calculate 2 x A which is 2 x 12 = 24

Finally, we get the answer by S - 2A = 1921.68 -24 = 1897.68

I enjoyed solving this question :)

Answer 2

Both bases are circles with an isosceles triangle cut out of them.

1. The areas of the circles (each with a radius of 6) is 36π
2. For isosceles triangles, the base is 8, and the altitude will hit the midpoint, creating two smaller right triangles, each with a leg of 4 and a hypotenuse of 5. Using Pythagorean Theorem, we get the altitude is 3. Using that with the area formula: 1/2(8)(3)=12
3. So base areas are each 36π-12

The cylinder has a lateral area of circumference x height = 12π(45) = 540π

The surface area is the sum of all areas, so 540π + 36π-12 + 36π-12 = 612π-24

Answer 3

Top: 36π - 12.5

Bottom: 36π - 12.5

Side: (12)(45)π

Interior: 5(45) + 5(45) + 8(45)