Surface Area of Solids worksheet with diagrams and calculations.

I will solve this worksheet problem by calculating the surface area for each of the 10 geometric solids shown. The instructions state to "Round to the nearest tenth."

Let me start by examining each figure one by one, identifying its shape, and then calculating its surface area using the appropriate formulas.

First, let's look at Figure 1:
This is a sphere with a diameter labeled as 12. To find the surface area of a sphere, I use the formula $A = 4\pi r^2$, where $r$ is the radius. Since the diameter is 12, the radius is 6.

Calculating:
$A = 4 \pi (6)^2 = 4 \pi (36) = 144\pi$
Using $\pi \approx 3.14159$, we get $A \approx 144 \times 3.14159 \approx 452.389$. Rounded to the nearest tenth, this is 452.4.

---

Now, let's examine Figure 2:
This appears to be a regular octahedron, which is a polyhedron with 8 equilateral triangular faces. The side length is given as 7. The surface area of a regular octahedron is $A = 2\sqrt{3} s^2$, where $s$ is the side length.

Calculating:
$A = 2\sqrt{3} (7)^2 = 2\sqrt{3} (49) = 98\sqrt{3}$
Using $\sqrt{3} \approx 1.73205$, we get $A \approx 98 \times 1.73205 \approx 169.741$. Rounded to the nearest tenth, this is 169.7.

---

Figure 3:
This is a rectangular prism (or cuboid). The dimensions are given as 10 (length), 5 (width), and 6 (height). The surface area of a rectangular prism is $A = 2(lw + lh + wh)$.

Calculating:
$A = 2[(10 \times 5) + (10 \times 6) + (5 \times 6)] = 2[50 + 60 + 30] = 2[140] = 280$

The surface area is 280.0.

---

Figure 4:
This is a cone. The radius ($r$) is 5, and the slant height ($l$) is 13. The surface area of a cone is $A = \pi r^2 + \pi r l$, which includes the base area and the lateral surface area.

Calculating:
$A = \pi (5)^2 + \pi (5)(13) = 25\pi + 65\pi = 90\pi$
Using $\pi \approx 3.14159$, we get $A \approx 90 \times 3.14159 \approx 282.743$. Rounded to the nearest tenth, this is 282.7.

---

Figure 5:
This is a square pyramid. The base is a square with side length 8, and the slant height of the triangular faces is 10. The surface area is the sum of the base area and the areas of the four triangular faces.

Base area: $8 \times 8 = 64$
Area of one triangular face: $(1/2) \times 8 \times 10 = 40$
Total area of four triangular faces: $4 \times 40 = 160$
Total surface area: $64 + 160 = 224$

The surface area is 224.0.

---

Figure 6:
This is a triangular pyramid (tetrahedron) with all edges equal to 8. This is a regular tetrahedron. The surface area is the sum of the areas of four equilateral triangles.

The area of one equilateral triangle with side $s$ is $(\sqrt{3}/4) s^2$.
So, total surface area $A = 4 \times (\sqrt{3}/4) s^2 = \sqrt{3} s^2$.

Calculating:
$A = \sqrt{3} (8)^2 = \sqrt{3} (64) = 64\sqrt{3}$
Using $\sqrt{3} \approx 1.73205$, we get $A \approx 64 \times 1.73205 \approx 110.851$. Rounded to the nearest tenth, this is 110.9.

---

Figure 7:
This is a triangular prism. The bases are right triangles with legs 6 and 8, so the hypotenuse is 10 (by the Pythagorean theorem). The height (length) of the prism is 12.

Surface area = 2 * (area of triangular base) + (perimeter of base * height of prism)

Area of one triangular base: $(1/2) \times 6 \times 8 = 24$
Perimeter of base: $6 + 8 + 10 = 24$
Lateral surface area: $24 \times 12 = 288$
Total surface area: $2 \times 24 + 288 = 48 + 288 = 336$

The surface area is 336.0.

---

Figure 8:
This is a trapezoidal prism. The bases of the trapezoid are 10 and 6, and the height of the trapezoid is 4. The length (height) of the prism is 8.

First, I need to find the area of the trapezoidal base: $A_{base} = (1/2) \times (b_1 + b_2) \times h = (1/2) \times (10 + 6) \times 4 = (1/2) \times 16 \times 4 = 32$

The perimeter of the trapezoid: I have the two parallel sides (10 and 6) and the height (4). I need to find the lengths of the non-parallel sides. Since it's not specified, I will assume it's an isosceles trapezoid, so the non-parallel sides are equal. Each non-parallel side can be found using the Pythagorean theorem on the right triangle formed by the height and half the difference of the bases: $(10 - 6)/2 = 2$. So, each non-parallel side is $\sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \approx 4.472$.

Perimeter of base: $10 + 6 + 2 \times 4.472 \approx 16 + 8.944 = 24.944$
Lateral surface area: $24.944 \times 8 \approx 199.552$
Total surface area: $2 \times 32 + 199.552 = 64 + 199.552 = 263.552$. Rounded to the nearest tenth, this is 263.6.

---

Figure 9:
This is a cylinder. The diameter is 12, so the radius is 6. The height is 10. The surface area of a cylinder is $A = 2\pi r^2 + 2\pi r h$ (two circular bases plus the lateral surface).

Calculating:
$A = 2\pi (6)^2 + 2\pi (6)(10) = 2\pi (36) + 2\pi (60) = 72\pi + 120\pi = 192\pi$
Using $\pi \approx 3.14159$, we get $A \approx 192 \times 3.14159 \approx 603.186$. Rounded to the nearest tenth, this is 603.2.

---

Figure 10:
This is a triangular prism. The base is a right triangle with legs 3 and 4, so the hypotenuse is 5. The height (length) of the prism is 10.

Surface area = 2 * (area of triangular base) + (perimeter of base * height of prism)

Area of one triangular base: $(1/2) \times 3 \times 4 = 6$
Perimeter of base: $3 + 4 + 5 = 12$
Lateral surface area: $12 \times 10 = 120$
Total surface area: $2 \times 6 + 120 = 12 + 120 = 132$

The surface area is 132.0.

---

Here is the final answer for each figure:

1. 452.4
2. 169.7
3. 280.0
4. 282.7
5. 224.0
6. 110.9
7. 336.0
8. 263.6
9. 603.2
10. 132.0