Question

2. The total surface area of a square based pyramid is 900 sq. cm. If the lateral surface area of the pyramid is three times the base area, find the length of the base. Also find the volume of the pyramid.​

Answer 1

Length of the base = 15 cm

Volume = 1125√2 cm³ = 1590.99 cm³ (2 d.p.)

Explanation:

`\boxed{\begin{minipage}{6.6cm}\underline{Lateral Surface Area of a square pyramid}\\\\$LSA=a √(a^2+4h^2)$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the base edge \\\phantom{ww}$\bullet$ $h$ is the height\\\end{minipage}}`

`\boxed{\begin{minipage}{6.6cm}\underline{Base Area of a square pyramid}\\\\$BA=a^2$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the base edge \\\end{minipage}}`

If the lateral surface area (LSA) of the pyramid is three times the base area (BA):

`\implies a √(a^2+4h^2)=3a^2`

Rearrange the equation to isolate h²:

`\implies √(a^2+4h^2)=3a`

`\implies (√(a^2+4h^2))^2=(3a)^2`

`\implies a^2+4h^2=9a^2`

`\implies 4h^2=8a^2`

`\implies h^2=2a^2`

`\boxed{\begin{minipage}{6.6cm}\underline{Total Surface Area of a square pyramid}\\\\$SA=a^2+a √(a^2+4h^2)$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the base edge \\\phantom{ww}$\bullet$ $h$ is the height\\\end{minipage}}`

Given:

• SA = 900 cm²
• h² = 2a²

Substitute these values into the formula for the total surface area and solve for a:

`\implies a^2+a√(a^2+4(2a^2))=900`

`\implies a^2+a√(a^2+8a^2)=900`

`\implies a^2+a√(9a^2)=900`

`\implies a^2+a(3a)=900`

`\implies a^2+3a^2=900`

`\implies 4a^2=900`

`\implies a^2=225`

`\implies a=15`

Therefore, the length of the base of the square based pyramid is:

• a = 15 cm

To find the height (h), substitute the found value of a into the formula for h² and solve for h:

`\implies h^2=2(15)^2`

`\implies √(h^2)=√(2(15)^2)`

`\implies h=√(2)√((15)^2)`

`\implies h=15√(2)`

Therefore, the height of the square based pyramid is:

• h = 15√2 cm

`\boxed{\begin{minipage}{5cm}\underline{Volume of a square pyramid}\\\\$V=(1)/(3)a^2h$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the base edge \\\phantom{ww}$\bullet$ $h$ is the height\\\end{minipage}}`

Given:

• a = 15
• h = 15√2

Substitute the values into the formula and solve for volume:

`\implies V=(1)/(3)(15)^2 \cdot 15√(2)`

`\implies V=(225)/(3) \cdot 15√(2)`

`\implies V=75\cdot 15√(2)`

`\implies V=1125√(2)`

`\implies V=1590.99`

Therefore, the volume of the square based pyramid is:

• 1590.99 cm³ (2 d.p.)