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Calculate the total area of the figure. Calculate the volume of the figure.

Mathematics

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2 parts
1

144 cm3144 \text{ cm}^3

2

258.53 cm2258.53 \text{ cm}^2

Step-by-step solution

7 steps

Here's how to calculate the volume and total surface area of the triangular prism:

Part 1· 3 steps

Calculate the Volume

  1. Identify the dimensions of the triangular base.

    The base of the triangle is b=3 cmb = 3 \text{ cm}.
    The height of the triangle is h=8 cmh = 8 \text{ cm}.

  2. Calculate the area of the triangular base (AbA_b).

    Ab=12×b×hA_b = \frac{1}{2} \times b \times h Ab=12×3 cm×8 cmA_b = \frac{1}{2} \times 3 \text{ cm} \times 8 \text{ cm} Ab=12 cm2A_b = 12 \text{ cm}^2

  3. Calculate the volume of the prism (VV).

    The length (height) of the prism is L=12 cmL = 12 \text{ cm}.

    V=Ab×LV = A_b \times L V=12 cm2×12 cmV = 12 \text{ cm}^2 \times 12 \text{ cm} V=144 cm3V = \boxed{144 \text{ cm}^3}

Part 2

258.53 cm2258.53 \text{ cm}^2

Calculate the Total Surface Area

  1. Calculate the area of the two triangular bases.

    Area of two bases = 2×Ab=2×12 cm2=24 cm22 \times A_b = 2 \times 12 \text{ cm}^2 = 24 \text{ cm}^2.

  2. Determine the lengths of all sides of the triangular base.

    The two given sides are 3 cm3 \text{ cm} and 8 cm8 \text{ cm}. Since it's a right-angled triangle (implied by the height measurement), we use the Pythagorean theorem to find the hypotenuse (cc).

    c=32+82c = \sqrt{3^2 + 8^2} c=9+64c = \sqrt{9 + 64} c=73 cmc = \sqrt{73} \text{ cm}

  3. Calculate the area of the three rectangular faces.

    The length of the prism is 12 cm12 \text{ cm}.

    • Area of the first rectangle (bottom) = 3 cm×12 cm=36 cm23 \text{ cm} \times 12 \text{ cm} = 36 \text{ cm}^2.
    • Area of the second rectangle (back) = 8 cm×12 cm=96 cm28 \text{ cm} \times 12 \text{ cm} = 96 \text{ cm}^2.
    • Area of the third rectangle (slanted) = 73 cm×12 cm=1273 cm2\sqrt{73} \text{ cm} \times 12 \text{ cm} = 12\sqrt{73} \text{ cm}^2.

  4. Calculate the total surface area (TSA).

    TSA=Area of two bases+Area of first rectangle+Area of second rectangle+Area of third rectangleTSA = \text{Area of two bases} + \text{Area of first rectangle} + \text{Area of second rectangle} + \text{Area of third rectangle} TSA=24 cm2+36 cm2+96 cm2+1273 cm2TSA = 24 \text{ cm}^2 + 36 \text{ cm}^2 + 96 \text{ cm}^2 + 12\sqrt{73} \text{ cm}^2 TSA=(24+36+96) cm2+1273 cm2TSA = (24 + 36 + 96) \text{ cm}^2 + 12\sqrt{73} \text{ cm}^2 TSA=(156+1273) cm2TSA = (156 + 12\sqrt{73}) \text{ cm}^2

    Using 73≈8.544\sqrt{73} \approx 8.544:

    TSA≈156+12×8.544 cm2TSA \approx 156 + 12 \times 8.544 \text{ cm}^2 TSA≈156+102.528 cm2TSA \approx 156 + 102.528 \text{ cm}^2 TSA≈258.53 cm2TSA \approx \boxed{258.53 \text{ cm}^2}
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