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Calculate the volume and total surface area of a square-based pyramid with height 24 cm and base side length 14 cm.

Mathematics

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3 parts
a

V=13a2hV = \frac{1}{3} a^2 h

b

896 cm2896 \text{ cm}^2

c

1568 cm31568 \text{ cm}^3

Step-by-step solution

4 steps

Here's the solution to question number 5:

Given:
Vertical height of a square-based pyramid, h=24h = 24 cm
Length of one side of base, a=14a = 14 cm

Part a

The formula to find the volume of a square-based pyramid is

V=13×base area×heightV = \frac{1}{3} \times \text{base area} \times \text{height}

Since the base is a square with side length aa, the base area is a2a^2.
So, the formula is:

V=13a2h\boxed{V = \frac{1}{3} a^2 h}

Part b

896 cm2896 \text{ cm}^2

To find the total surface area of the pyramid, we first need to calculate the slant height (ll).

  1. Calculate the slant height (ll).

    The slant height can be found using the Pythagorean theorem:

    l=h2+(a2)2l = \sqrt{h^2 + \left(\frac{a}{2}\right)^2} l=242+(142)2l = \sqrt{24^2 + \left(\frac{14}{2}\right)^2} l=242+72l = \sqrt{24^2 + 7^2} l=576+49l = \sqrt{576 + 49} l=625l = \sqrt{625} l=25 cml = 25 \text{ cm}

  2. Calculate the total surface area (TSA).

    The total surface area is the sum of the base area and the lateral surface area.

    TSA=Base Area+Lateral Surface Area\text{TSA} = \text{Base Area} + \text{Lateral Surface Area} TSA=a2+2al\text{TSA} = a^2 + 2al TSA=142+2×14×25\text{TSA} = 14^2 + 2 \times 14 \times 25 TSA=196+700\text{TSA} = 196 + 700 TSA=896 cm2\text{TSA} = \boxed{896 \text{ cm}^2}

Part c

1568 cm31568 \text{ cm}^3

To find the volume of the pyramid

  1. Use the formula for the volume of a square-based pyramid.

    V=13a2hV = \frac{1}{3} a^2 h

  2. Substitute the given values and calculate.

    V=13×142×24V = \frac{1}{3} \times 14^2 \times 24 V=13×196×24V = \frac{1}{3} \times 196 \times 24 V=196×8V = 196 \times 8 V=1568 cm3V = \boxed{1568 \text{ cm}^3}
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