worksheet-surface-area-and- ... - Free Printable

The task involves calculating the surface area and volume of various 3D geometric figures. Below, I will explain how to solve these problems step by step for each figure.

---

1. Rectangular Prism (Top Left)

Dimensions: Length = 7 mm, Width = 17 mm, Height = 15 mm

#### Surface Area:
The formula for the surface area of a rectangular prism is:
\[
\text{Surface Area} = 2(lw + lh + wh)
\]
Substitute the values:
\[
\text{Surface Area} = 2(7 \times 17 + 7 \times 15 + 17 \times 15)
\]
\[
= 2(119 + 105 + 255)
\]
\[
= 2 \times 489 = 978 \, \text{mm}^2
\]

#### Volume:
The formula for the volume of a rectangular prism is:
\[
\text{Volume} = l \times w \times h
\]
Substitute the values:
\[
\text{Volume} = 7 \times 17 \times 15 = 1785 \, \text{mm}^3
\]

Answer:
\[
\text{Surface Area} = 978 \, \text{mm}^2, \quad \text{Volume} = 1785 \, \text{mm}^3
\]

---

2. Square Pyramid (Top Middle)

Base Side = 6 yd, Slant Height = 9 yd

#### Surface Area:
The surface area of a square pyramid consists of the base area and the area of the four triangular faces.
\[
\text{Base Area} = s^2 = 6^2 = 36 \, \text{yd}^2
\]
The area of one triangular face is:
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 6 \times 9 = 27 \, \text{yd}^2
\]
Since there are 4 triangular faces:
\[
\text{Total area of triangles} = 4 \times 27 = 108 \, \text{yd}^2
\]
Thus, the total surface area is:
\[
\text{Surface Area} = \text{Base Area} + \text{Total area of triangles} = 36 + 108 = 144 \, \text{yd}^2
\]

#### Volume:
The formula for the volume of a pyramid is:
\[
\text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
We need the height of the pyramid. Using the Pythagorean theorem in the right triangle formed by the height, half the base side, and the slant height:
\[
\text{Height} = \sqrt{\text{Slant Height}^2 - \left(\frac{\text{Base Side}}{2}\right)^2} = \sqrt{9^2 - 3^2} = \sqrt{81 - 9} = \sqrt{72} = 6\sqrt{2} \, \text{yd}
\]
Now, calculate the volume:
\[
\text{Volume} = \frac{1}{3} \times 36 \times 6\sqrt{2} = 72\sqrt{2} \approx 101.82 \, \text{yd}^3
\]

Answer:
\[
\text{Surface Area} = 144 \, \text{yd}^2, \quad \text{Volume} \approx 101.82 \, \text{yd}^3
\]

---

3. Cube (Top Right)

Side = 6 ft

#### Surface Area:
The formula for the surface area of a cube is:
\[
\text{Surface Area} = 6s^2
\]
Substitute the value:
\[
\text{Surface Area} = 6 \times 6^2 = 6 \times 36 = 216 \, \text{ft}^2
\]

#### Volume:
The formula for the volume of a cube is:
\[
\text{Volume} = s^3
\]
Substitute the value:
\[
\text{Volume} = 6^3 = 216 \, \text{ft}^3
\]

Answer:
\[
\text{Surface Area} = 216 \, \text{ft}^2, \quad \text{Volume} = 216 \, \text{ft}^3
\]

---

4. Triangular Pyramid (Middle Left)

Base Side = 6 cm, Height = 8 cm, Slant Height = 10 cm

#### Surface Area:
The surface area consists of the base area and the areas of the three triangular faces.
- Base Area: The base is an equilateral triangle with side 6 cm.
\[
\text{Base Area} = \frac{\sqrt{3}}{4} \times s^2 = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \approx 15.59 \, \text{cm}^2
\]
- Lateral Faces: Each lateral face is a triangle with base 6 cm and slant height 10 cm.
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 6 \times 10 = 30 \, \text{cm}^2
\]
Since there are 3 triangular faces:
\[
\text{Total area of triangles} = 3 \times 30 = 90 \, \text{cm}^2
\]
Thus, the total surface area is:
\[
\text{Surface Area} = \text{Base Area} + \text{Total area of triangles} = 15.59 + 90 = 105.59 \, \text{cm}^2
\]

#### Volume:
The formula for the volume of a pyramid is:
\[
\text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Substitute the values:
\[
\text{Volume} = \frac{1}{3} \times 15.59 \times 8 \approx \frac{1}{3} \times 124.72 \approx 41.57 \, \text{cm}^3
\]

Answer:
\[
\text{Surface Area} \approx 105.59 \, \text{cm}^2, \quad \text{Volume} \approx 41.57 \, \text{cm}^3
\]

---

5. Triangular Pyramid (Middle Middle)

Base Side = 8 in, Height = 20 in

#### Surface Area:
The surface area consists of the base area and the areas of the three triangular faces.
- Base Area: The base is an equilateral triangle with side 8 in.
\[
\text{Base Area} = \frac{\sqrt{3}}{4} \times s^2 = \frac{\sqrt{3}}{4} \times 8^2 = \frac{\sqrt{3}}{4} \times 64 = 16\sqrt{3} \approx 27.71 \, \text{in}^2
\]
- Lateral Faces: We need the slant height of the triangular faces. Using the Pythagorean theorem in the right triangle formed by the height of the pyramid, the apothem of the base, and the slant height:
\[
\text{Apothem} = \frac{\sqrt{3}}{2} \times \text{side} = \frac{\sqrt{3}}{2} \times 8 = 4\sqrt{3} \, \text{in}
\]
\[
\text{Slant Height} = \sqrt{\text{Height}^2 + \text{Apothem}^2} = \sqrt{20^2 + (4\sqrt{3})^2} = \sqrt{400 + 48} = \sqrt{448} = 8\sqrt{7} \, \text{in}
\]
The area of one triangular face is:
\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 8 \times 8\sqrt{7} = 32\sqrt{7} \approx 81.28 \, \text{in}^2
\]
Since there are 3 triangular faces:
\[
\text{Total area of triangles} = 3 \times 81.28 = 243.84 \, \text{in}^2
\]
Thus, the total surface area is:
\[
\text{Surface Area} = \text{Base Area} + \text{Total area of triangles} = 27.71 + 243.84 = 271.55 \, \text{in}^2
\]

#### Volume:
The formula for the volume of a pyramid is:
\[
\text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Substitute the values:
\[
\text{Volume} = \frac{1}{3} \times 27.71 \times 20 \approx \frac{1}{3} \times 554.2 \approx 184.73 \, \text{in}^3
\]

Answer:
\[
\text{Surface Area} \approx 271.55 \, \text{in}^2, \quad \text{Volume} \approx 184.73 \, \text{in}^3
\]

---

6. Cylinder (Middle Right)

Radius = 3.3 cm, Height = 7 cm

#### Surface Area:
The formula for the surface area of a cylinder is:
\[
\text{Surface Area} = 2\pi r(h + r)
\]
Substitute the values:
\[
\text{Surface Area} = 2\pi \times 3.3 \times (7 + 3.3) = 2\pi \times 3.3 \times 10.3 = 68.34\pi \approx 214.74 \, \text{cm}^2
\]

#### Volume:
The formula for the volume of a cylinder is:
\[
\text{Volume} = \pi r^2 h
\]
Substitute the values:
\[
\text{Volume} = \pi \times 3.3^2 \times 7 = \pi \times 10.89 \times 7 = 76.23\pi \approx 239.47 \, \text{cm}^3
\]

Answer:
\[
\text{Surface Area} \approx 214.74 \, \text{cm}^2, \quad \text{Volume} \approx 239.47 \, \text{cm}^3
\]

---

7. Cylinder (Bottom Left)

Radius = 7 mm, Height = 15 mm

#### Surface Area:
The formula for the surface area of a cylinder is:
\[
\text{Surface Area} = 2\pi r(h + r)
\]
Substitute the values:
\[
\text{Surface Area} = 2\pi \times 7 \times (15 + 7) = 2\pi \times 7 \times 22 = 308\pi \approx 967.69 \, \text{mm}^2
\]

#### Volume:
The formula for the volume of a cylinder is:
\[
\text{Volume} = \pi r^2 h
\]
Substitute the values:
\[
\text{Volume} = \pi \times 7^2 \times 15 = \pi \times 49 \times 15 = 735\pi \approx 2307.95 \, \text{mm}^3
\]

Answer:
\[
\text{Surface Area} \approx 967.69 \, \text{mm}^2, \quad \text{Volume} \approx 2307.95 \, \text{mm}^3
\]

---

8. Sphere (Bottom Middle)

Radius = 13 in

#### Surface Area:
The formula for the surface area of a sphere is:
\[
\text{Surface Area} = 4\pi r^2
\]
Substitute the value:
\[
\text{Surface Area} = 4\pi \times 13^2 = 4\pi \times 169 = 676\pi \approx 2123.72 \, \text{in}^2
\]

#### Volume:
The formula for the volume of a sphere is:
\[
\text{Volume} = \frac{4}{3}\pi r^3
\]
Substitute the value:
\[
\text{Volume} = \frac{4}{3}\pi \times 13^3 = \frac{4}{3}\pi \times 2197 = \frac{8788}{3}\pi \approx 9203.77 \, \text{in}^3
\]

Answer:
\[
\text{Surface Area} \approx 2123.72 \, \text{in}^2, \quad \text{Volume} \approx 9203.77 \, \text{in}^3
\]

---

9. Triangular Prism (Bottom Right)

Base Side = 2 yd, Height = 11 yd, Slant Height = 5 yd

#### Surface Area:
The surface area consists of the two triangular bases and the three rectangular lateral faces.
- Base Area: The base is an equilateral triangle with side 2 yd.
\[
\text{Base Area} = \frac{\sqrt{3}}{4} \times s^2 = \frac{\sqrt{3}}{4} \times 2^2 = \frac{\sqrt{3}}{4} \times 4 = \sqrt{3} \approx 1.73 \, \text{yd}^2
\]
- Lateral Faces: There are three rectangular faces, each with dimensions 2 yd (base side) and 11 yd (height).
\[
\text{Area of one rectangle} = 2 \times 11 = 22 \, \text{yd}^2
\]
Since there are 3 rectangles:
\[
\text{Total area of rectangles} = 3 \times 22 = 66 \, \text{yd}^2
\]
Thus, the total surface area is:
\[
\text{Surface Area} = 2 \times \text{Base Area} + \text{Total area of rectangles} = 2 \times 1.73 + 66 = 3.46 + 66 = 69.46 \, \text{yd}^2
\]

#### Volume:
The formula for the volume of a prism is:
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
Substitute the values:
\[
\text{Volume} = 1.73 \times 11 \approx 19.03 \, \text{yd}^3
\]

Answer:
\[
\text{Surface Area} \approx 69.46 \, \text{yd}^2, \quad \text{Volume} \approx 19.03 \, \text{yd}^3
\]

---

10. Rectangular Prism (Bottom Left)

Length = 9 ft, Width = 7 ft, Height = 7 ft

#### Surface Area:
The formula for the surface area of a rectangular prism is:
\[
\text{Surface Area} = 2(lw + lh + wh)
\]
Substitute the values:
\[
\text{Surface Area} = 2(9 \times 7 + 9 \times 7 + 7 \times 7)
\]
\[
= 2(63 + 63 + 49)
\]
\[
= 2 \times 175 = 350 \, \text{ft}^2
\]

#### Volume:
The formula for the volume of a rectangular prism is:
\[
\text{Volume} = l \times w \times h
\]
Substitute the values:
\[
\text{Volume} = 9 \times 7 \times 7 = 441 \, \text{ft}^3
\]

Answer:
\[
\text{Surface Area} = 350 \, \text{ft}^2, \quad \text{Volume} = 441 \, \text{ft}^3
\]

---

11. Hexagonal Prism (Bottom Middle)

Base Side = 5 mm, Height = 14 mm

#### Surface Area:
The surface area consists of the two hexagonal bases and the six rectangular lateral faces.
- Base Area: The area of a regular hexagon with side \( s \) is:
\[
\text{Base Area} = \frac{3\sqrt{3}}{2} s^2
\]
Substitute the value:
\[
\text{Base Area} = \frac{3\sqrt{3}}{2} \times 5^2 = \frac{3\sqrt{3}}{2} \times 25 = \frac{75\sqrt{3}}{2} \approx 64.95 \, \text{mm}^2
\]
- Lateral Faces: There are six rectangular faces, each with dimensions 5 mm (base side) and 14 mm (height).
\[
\text{Area of one rectangle} = 5 \times 14 = 70 \, \text{mm}^2
\]
Since there are 6 rectangles:
\[
\text{Total area of rectangles} = 6 \times 70 = 420 \, \text{mm}^2
\]
Thus, the total surface area is:
\[
\text{Surface Area} = 2 \times \text{Base Area} + \text{Total area of rectangles} = 2 \times 64.95 + 420 = 129.9 + 420 = 549.9 \, \text{mm}^2
\]

#### Volume:
The formula for the volume of a prism is:
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
Substitute the values:
\[
\text{Volume} = 64.95 \times 14 \approx 909.3 \, \text{mm}^3
\]

Answer:
\[
\text{Surface Area} \approx 549.9 \, \text{mm}^2, \quad \text{Volume} \approx 909.3 \, \text{mm}^3
\]

---

12. Trapezoidal Prism (Bottom Right)

Base 1 = 7 yd, Base 2 = 17 yd, Height of trapezoid = 5 yd, Prism height = 7 yd

#### Surface Area:
The surface area consists of the two trapezoidal bases and the four rectangular lateral faces.
- Base Area: The area of a trapezoid is:
\[
\text{Base Area} = \frac{1}{2} \times (b_1 + b_2) \times h
\]
Substitute the values:
\[
\text{Base Area} = \frac{1}{2} \times (7 + 17) \times 5 = \frac{1}{2} \times 24 \times 5 = 60 \, \text{yd}^2
\]
- Lateral Faces: There are four rectangular faces. Two have dimensions 7 yd (short base) and 7 yd (prism height), and two have dimensions 17 yd (long base) and 7 yd (prism height).
\[
\text{Area of short base rectangles} = 2 \times (7 \times 7) = 2 \times 49 = 98 \, \text{yd}^2
\]
\[
\text{Area of long base rectangles} = 2 \times (17 \times 7) = 2 \times 119 = 238 \, \text{yd}^2
\]
Thus, the total surface area is:
\[
\text{Surface Area} = 2 \times \text{Base Area} + \text{Total area of rectangles} = 2 \times 60 + 98 + 238 = 120 + 98 + 238 = 456 \, \text{yd}^2
\]

#### Volume:
The formula for the volume of a prism is:
\[
\text{Volume} = \text{Base Area} \times \text{Height}
\]
Substitute the values:
\[
\text{Volume} = 60 \times 7 = 420 \, \text{yd}^3
\]

Answer:
\[
\text{Surface Area} = 456 \, \text{yd}^2, \quad \text{Volume} = 420 \, \text{yd}^3
\]

---

Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{Rectangular Prism (Top Left)} & \text{Surface Area} = 978 \, \text{mm}^2, \, \text{Volume} = 1785 \, \text{mm}^3 \\
\text{Square Pyramid (Top Middle)} & \text{Surface Area} = 144 \, \text{yd}^2, \, \text{Volume} \approx 101.82 \, \text{yd}^3 \\
\text{Cube (Top Right)} & \text{Surface Area} = 216 \, \text{ft}^2, \, \text{Volume} = 216 \, \text{ft}^3 \\
\text{Triangular Pyramid (Middle Left)} & \text{Surface Area} \approx 105.59 \, \text{cm}^2, \, \text{Volume} \approx 41.57 \, \text{cm}^3 \\
\text{Triangular Pyramid (Middle Middle)} & \text{Surface Area} \approx 271.55 \, \text{in}^2, \, \text{Volume} \approx 184.73 \, \text{in}^3 \\
\text{Cylinder (Middle Right)} & \text{Surface Area} \approx 214.74 \, \text{cm}^2, \, \text{Volume} \approx 239.47 \, \text{cm}^3 \\
\text{Cylinder (Bottom Left)} & \text{Surface Area} \approx 967.69 \, \text{mm}^2, \, \text{Volume} \approx 2307.95 \, \text{mm}^3 \\
\text{Sphere (Bottom Middle)} & \text{Surface Area} \approx 2123.72 \, \text{in}^2, \, \text{Volume} \approx 9203.77 \, \text{in}^3 \\
\text{Triangular Prism (Bottom Right)} & \text{Surface Area} \approx 69.46 \, \text{yd}^2, \, \text{Volume} \approx 19.03 \, \text{yd}^3 \\
\text{Rectangular Prism (Bottom Left)} & \text{Surface Area} = 350 \, \text{ft}^2, \, \text{Volume} = 441 \, \text{ft}^3 \\
\text{Hexagonal Prism (Bottom Middle)} & \text{Surface Area} \approx 549.9 \, \text{mm}^2, \, \text{Volume} \approx 909.3 \, \text{mm}^3 \\
\text{Trapezoidal Prism (Bottom Right)} & \text{Surface Area} = 456 \, \text{yd}^2, \, \text{Volume} = 420 \, \text{yd}^3 \\
\end{array}
}
\]