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Calculate the volumes of the following shapes: a) Rectangular prism, b) Cylinder, c) Triangular prism.

Mathematics

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4 parts
a

120 cm3\text{120 cm}^3

b

769.69 cm3\text{769.69 cm}^3

c

480 cm3\text{480 cm}^3

d

48 cm3\text{48 cm}^3

Step-by-step solution

14 steps

Here are the volumes of the given shapes:

Part a· 3 steps

This is a rectangular prism.

  1. Identify the dimensions.

    The length l=8l = 8 cm, width w=5w = 5 cm, and height h=3h = 3 cm.

  2. Apply the formula for the volume of a rectangular prism.

    The volume V=l×w×hV = l \times w \times h.

    Va=8 cm×5 cm×3 cmV_a = 8 \text{ cm} \times 5 \text{ cm} \times 3 \text{ cm}

  3. Calculate the volume.

    Va=40 cm2×3 cmV_a = 40 \text{ cm}^2 \times 3 \text{ cm} Va=120 cm3V_a = \boxed{\text{120 cm}^3}

Part b

769.69 cm3\text{769.69 cm}^3

This is a cylinder.

  1. Identify the dimensions.

    The diameter d=14d = 14 cm, so the radius r=142=7r = \frac{14}{2} = 7 cm. The height h=5h = 5 cm.

  2. Apply the formula for the volume of a cylinder.

    The volume V=πr2hV = \pi r^2 h.

    Vb=π×(7 cm)2×5 cmV_b = \pi \times (7 \text{ cm})^2 \times 5 \text{ cm}

  3. Calculate the volume.

    Vb=π×49 cm2×5 cmV_b = \pi \times 49 \text{ cm}^2 \times 5 \text{ cm} Vb=245π cm3V_b = 245\pi \text{ cm}^3

    Using π≈3.14159\pi \approx 3.14159:

    Vb≈245×3.14159 cm3V_b \approx 245 \times 3.14159 \text{ cm}^3 Vb≈769.69 cm3V_b \approx \boxed{\text{769.69 cm}^3}

Part c

480 cm3\text{480 cm}^3

This is a triangular prism.

  1. Identify the dimensions of the triangular base and the prism's length.

    The base of the triangle bt=8b_t = 8 cm, the height of the triangle ht=6h_t = 6 cm. The length of the prism L=20L = 20 cm.

  2. Calculate the area of the triangular base.

    The area of the base A=12×bt×htA = \frac{1}{2} \times b_t \times h_t.

    Ac=12×8 cm×6 cmA_c = \frac{1}{2} \times 8 \text{ cm} \times 6 \text{ cm} Ac=24 cm2A_c = 24 \text{ cm}^2

  3. Apply the formula for the volume of a prism.

    The volume V=A×LV = A \times L.

    Vc=24 cm2×20 cmV_c = 24 \text{ cm}^2 \times 20 \text{ cm}

  4. Calculate the volume.

    Vc=480 cm3V_c = \boxed{\text{480 cm}^3}

Part d

48 cm3\text{48 cm}^3

This is a square pyramid.

  1. Identify the dimensions of the square base and the pyramid's height.

    The side length of the square base s=6s = 6 cm. The height of the pyramid hp=4h_p = 4 cm.

  2. Calculate the area of the square base.

    The area of the base A=s2A = s^2.

    Ad=(6 cm)2A_d = (6 \text{ cm})^2 Ad=36 cm2A_d = 36 \text{ cm}^2

  3. Apply the formula for the volume of a pyramid.

    The volume V=13×A×hpV = \frac{1}{3} \times A \times h_p.

    Vd=13×36 cm2×4 cmV_d = \frac{1}{3} \times 36 \text{ cm}^2 \times 4 \text{ cm}

  4. Calculate the volume.

    Vd=12 cm2×4 cmV_d = 12 \text{ cm}^2 \times 4 \text{ cm} Vd=48 cm3V_d = \boxed{\text{48 cm}^3}
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