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I. Find the L.S.A, T.S.A and volume of cuboid.

1. l = 10 cm, b = 8 cm and h = 5 cm.
2. l = 12.5 cm, b = 9 cm and h = 7.5 cm.

II. Find the L.S.A, T.S.A and volume of cube.

1. Edge of cube = 16cm
2. Edge of cube = 10.5cm

III.

1. The height of a right prism is 15cm. Its base is a triangle with sides 10cm, 17cm, 9cm. Find L.S.A, T.S.A and volume of a prism.
2. The height of a right prism is 12cm. Its base is an equilateral triangle with sides 6cm. Find L.S.A, T.S.A and volume of a prism.

Answer :

Let's solve these problems step by step. We'll go through each part one by one.

I. Cuboid Calculations

1. For a cuboid with

[tex]l = 10[/tex]

cm,

[tex]b = 8[/tex]

cm, and

[tex]h = 5[/tex]

cm:

  • Lateral Surface Area (L.S.A): The lateral surface area of a cuboid is given by [tex]2h(l + b)[/tex].
    [tex]\text{L.S.A} = 2 \times 5 \times (10 + 8) = 2 \times 5 \times 18 = 180 \text{ cm}^2[/tex]
  • Total Surface Area (T.S.A): The total surface area is given by [tex]2(lb + bh + hl)[/tex].
    [tex]\text{T.S.A} = 2(10 \times 8 + 8 \times 5 + 5 \times 10) = 2(80 + 40 + 50) = 2 \times 170 = 340 \text{ cm}^2[/tex]
  • Volume: The volume is given by [tex]l \times b \times h[/tex].
    [tex]\text{Volume} = 10 \times 8 \times 5 = 400 \text{ cm}^3[/tex]

2. For a cuboid with

[tex]l = 12.5[/tex]

cm,

[tex]b = 9[/tex]

cm, and

[tex]h = 7.5[/tex]

cm:

  • Lateral Surface Area (L.S.A):
    [tex]\text{L.S.A} = 2 \times 7.5 \times (12.5 + 9) = 2 \times 7.5 \times 21.5 = 322.5 \text{ cm}^2[/tex]
  • Total Surface Area (T.S.A):
    [tex]\text{T.S.A} = 2(12.5 \times 9 + 9 \times 7.5 + 7.5 \times 12.5) = 2(112.5 + 67.5 + 93.75) = 2 \times 273.75 = 547.5 \text{ cm}^2[/tex]
  • Volume:
    [tex]\text{Volume} = 12.5 \times 9 \times 7.5 = 843.75 \text{ cm}^3[/tex]

II. Cube Calculations

For a cube, all edges are the same, so the calculations are simplified.

1. For a cube with edge

[tex]s = 16[/tex]

cm:

  • Lateral Surface Area (L.S.A): [tex]4s^2[/tex]
    [tex]\text{L.S.A} = 4 \times 16^2 = 4 \times 256 = 1024 \text{ cm}^2[/tex]
  • Total Surface Area (T.S.A): [tex]6s^2[/tex]
    [tex]\text{T.S.A} = 6 \times 16^2 = 6 \times 256 = 1536 \text{ cm}^2[/tex]
  • Volume: [tex]s^3[/tex]
    [tex]\text{Volume} = 16^3 = 4096 \text{ cm}^3[/tex]

2. For a cube with edge

[tex]s = 10.5[/tex]

cm:

  • Lateral Surface Area (L.S.A):
    [tex]\text{L.S.A} = 4 \times 10.5^2 = 4 \times 110.25 = 441 \text{ cm}^2[/tex]
  • Total Surface Area (T.S.A):
    [tex]\text{T.S.A} = 6 \times 10.5^2 = 6 \times 110.25 = 661.5 \text{ cm}^2[/tex]
  • Volume:
    [tex]\text{Volume} = 10.5^3 = 1157.625 \text{ cm}^3[/tex]

III. Right Prism Calculations

1. For a prism with height = 15 cm and a triangular base with sides

[tex]10[/tex]

cm,

[tex]17[/tex]

cm, and

[tex]9[/tex]

cm:

  • To find the area of the triangular base, use Heron's formula. Let [tex]s[/tex] be the semi-perimeter: [tex]s = \frac{10 + 17 + 9}{2} = 18[/tex].
  • Area of base:
    [tex]A = \sqrt{s(s - a)(s - b)(s - c)}[/tex]
    [tex]A = \sqrt{18(18 - 10)(18 - 17)(18 - 9)} = \sqrt{18 \times 8 \times 1 \times 9} = \sqrt{1296} = 36 \text{ cm}^2[/tex]
  • Lateral Surface Area (L.S.A): [tex]ext{Perimeter of base} \times ext{Height} = (10 + 17 + 9) \times 15 = 36 \times 15 = 540 \text{ cm}^2[/tex].
  • Total Surface Area (T.S.A): [tex]ext{Lateral Surface Area} + 2 \times ext{Area of base} = 540 + 2 \times 36 = 612 \text{ cm}^2[/tex].
  • Volume: [tex]ext{Area of base} \times ext{Height} = 36 \times 15 = 540 \text{ cm}^3[/tex].

2. For a prism with height = 12 cm and an equilateral triangular base with side = 6 cm:

  • Area of base: For an equilateral triangle, [tex]A = \frac{\sqrt{3}}{4} a^2[/tex].
    [tex]A = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \text{ cm}^2[/tex]
  • Lateral Surface Area (L.S.A): [tex]ext{Perimeter of base} \times ext{Height} = 18 \times 12 = 216 \text{ cm}^2[/tex].
  • Total Surface Area (T.S.A): [tex]ext{Lateral Surface Area} + 2 \times ext{Area of base} = 216 + 2 \times 9\sqrt{3} \text{ cm}^2[/tex].
  • Volume: [tex]ext{Area of base} \times ext{Height} = 9\sqrt{3} \times 12 = 108\sqrt{3} \text{ cm}^3[/tex].

These calculations aim to give a comprehensive understanding of the geometric properties of cuboids, cubes, and right prisms. Let me know if you have any further questions!

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Rewritten by : Barada