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A square-based pyramid has a base side of 20 cm and a slant edge of 26 cm. How many total surfaces are there, and what is the total surface area?

Mathematics

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Given:
Length of a side of the base of a square-based pyramid (aa) = 20 cm
Length of slant edge (ee) = 26 cm

Part a

How many total surfaces are there in a square base pyramid? Write it.

A square-based pyramid has one square base and four triangular lateral faces.
Total number of surfaces = 1(base)+4(lateral faces)=51 (\text{base}) + 4 (\text{lateral faces}) = 5.
The final answer is 5 surfaces\boxed{5 \text{ surfaces}}.

Part b· 4 steps

Find the total surface area of the pyramid?

  1. Calculate the area of the base.

    The base is a square with side length a=20a = 20 cm.

    Area of base=a2=(20 cm)2=400 cm2\text{Area of base} = a^2 = (20 \text{ cm})^2 = 400 \text{ cm}^2

  2. Calculate the slant height (ll) of the pyramid.

    Consider a right-angled triangle formed by the slant edge (ee), half the base side (a2\frac{a}{2}), and the slant height (ll). The slant edge is the hypotenuse.

    e2=l2+(a2)2e^2 = l^2 + \left(\frac{a}{2}\right)^2 (26 cm)2=l2+(20 cm2)2(26 \text{ cm})^2 = l^2 + \left(\frac{20 \text{ cm}}{2}\right)^2 676 cm2=l2+(10 cm)2676 \text{ cm}^2 = l^2 + (10 \text{ cm})^2 676 cm2=l2+100 cm2676 \text{ cm}^2 = l^2 + 100 \text{ cm}^2 l2=676 cm2−100 cm2l^2 = 676 \text{ cm}^2 - 100 \text{ cm}^2 l2=576 cm2l^2 = 576 \text{ cm}^2 l=576 cm2=24 cml = \sqrt{576 \text{ cm}^2} = 24 \text{ cm}

  3. Calculate the lateral surface area of the pyramid.

    The lateral surface area is the sum of the areas of the four triangular faces.

    Lateral surface area=12×Perimeter of base×l\text{Lateral surface area} = \frac{1}{2} \times \text{Perimeter of base} \times l Perimeter of base=4a=4×20 cm=80 cm\text{Perimeter of base} = 4a = 4 \times 20 \text{ cm} = 80 \text{ cm} Lateral surface area=12×80 cm×24 cm\text{Lateral surface area} = \frac{1}{2} \times 80 \text{ cm} \times 24 \text{ cm} Lateral surface area=40 cm×24 cm=960 cm2\text{Lateral surface area} = 40 \text{ cm} \times 24 \text{ cm} = 960 \text{ cm}^2

  4. Calculate the total surface area.

    Total surface area=Area of base+Lateral surface area\text{Total surface area} = \text{Area of base} + \text{Lateral surface area} Total surface area=400 cm2+960 cm2=1360 cm2\text{Total surface area} = 400 \text{ cm}^2 + 960 \text{ cm}^2 = 1360 \text{ cm}^2

    The final answer is 1360 cm2\boxed{1360 \text{ cm}^2}.

Part c· 2 steps

Compare between the vertical height and slant height of the pyramid.

  1. Calculate the vertical height (hh) of the pyramid.

    Consider a right-angled triangle formed by the vertical height (hh), half the base side (a2\frac{a}{2}), and the slant height (ll) as the hypotenuse.

    l2=h2+(a2)2l^2 = h^2 + \left(\frac{a}{2}\right)^2

    We found l=24l = 24 cm and a2=10\frac{a}{2} = 10 cm.

    (24 cm)2=h2+(10 cm)2(24 \text{ cm})^2 = h^2 + (10 \text{ cm})^2 576 cm2=h2+100 cm2576 \text{ cm}^2 = h^2 + 100 \text{ cm}^2 h2=576 cm2−100 cm2h^2 = 576 \text{ cm}^2 - 100 \text{ cm}^2 h2=476 cm2h^2 = 476 \text{ cm}^2 h=476 cm2=4×119 cm=2119 cmh = \sqrt{476 \text{ cm}^2} = \sqrt{4 \times 119} \text{ cm} = 2\sqrt{119} \text{ cm}

    To compare, we can approximate 21192\sqrt{119}:

    119≈10.9087\sqrt{119} \approx 10.9087 h≈2×10.9087 cm≈21.8174 cmh \approx 2 \times 10.9087 \text{ cm} \approx 21.8174 \text{ cm}

  2. Compare the vertical height (hh) and slant height (ll).

    Vertical height h=2119h = 2\sqrt{119} cm ≈21.82\approx 21.82 cm
    Slant height l=24l = 24 cm
    Since 24>211924 > 2\sqrt{119}, the slant height is greater than the vertical height.
    The final answer is The slant height (24 cm) is greater than the vertical height (2119 cm ≈21.82 cm).\boxed{\text{The slant height (24 cm) is greater than the vertical height (} 2\sqrt{119} \text{ cm } \approx 21.82 \text{ cm}).}.

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