Surface Area of Pyramid

Question 1

Find the area of lateral and total surfaces of a square based pyramid if the length of side (a) = 8 cm and slant height (l) = 3 cm.

Solution:

Given information:

• Length of the side of base (a) = 8 cm
• Slant height (l) = 3 cm

Step 1: Calculate the area of base

We know that the area of base = $a^2$

$\text{Area of base} = (8 \text{ cm})^2 = 64 \text{ cm}^2$

Step 2: Calculate lateral surface area

Lateral surface area (LSA) = $2al$

$\text{LSA} = 2 \times 8 \times 3 = 48 \text{ cm}^2$

Step 3: Calculate total surface area

Total surface area = Area of base + Lateral surface area

$\text{Total surface area} = 64 \text{ cm}^2 + 48 \text{ cm}^2 = 112 \text{ cm}^2$

Final Answer:

Hence, the lateral surface area = 48 cm² and total surface area = 112 cm².

Verification:

• Base area: $a^2 = 8^2 = 64 \text{ cm}^2$
• Lateral surface area: $2al = 2 \times 8 \times 3 = 48 \text{ cm}^2$
• Total surface area: $64 + 48 = 112 \text{ cm}^2$ ✓

Formula explanation:

• Square base area: $\text{Area} = a^2$ where $a$ is the side length
• Lateral surface area: $\text{LSA} = \text{Perimeter of base} \times \frac{\text{slant height}}{2} = 4a \times \frac{l}{2} = 2al$
• Total surface area: $\text{TSA} = \text{Base area} + \text{LSA}$

Note: For a square-based pyramid, the lateral surface area consists of 4 triangular faces. Each triangular face has area = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times l$. So total LSA = $4 \times \frac{1}{2} \times a \times l = 2al$.

Question 2

Find the total surface area of a square based pyramid with the length of the side of base is 10 cm and the length of the edge is 13 cm.

Solution:

Given information:

• Length of the side of base (a) = 10 cm
• Length of the edge (e) = 13 cm
• Total surface area = ?

Step 1: Find the slant height (l)

We know that in a square-based pyramid:

$e^2 = l^2 + \left(\frac{a}{2}\right)^2$

Substituting the values:

$(13)^2 = l^2 + \left(\frac{10}{2}\right)^2$ $169 = l^2 + 25$ $169 - 25 = l^2$ $144 = l^2$ $l = 12 \text{ cm}$

∴ The slant height (l) = 12 cm

Step 2: Calculate the area of base

Area of base (A) = $a^2 = (10)^2 = 100 \text{ cm}^2$

Step 3: Calculate lateral surface area

Lateral surface area (LSA) = $2al = 2 \times 10 \times 12 = 240 \text{ cm}^2$

Step 4: Calculate total surface area

Total surface area = Area of base (A) + Lateral surface area

$\text{Total surface area} = 100 + 240 = 340 \text{ cm}^2$

Final Answer:

Hence, the total surface area of the pyramid is 340 cm².

Verification:

• Edge length check: $e^2 = l^2 + \left(\frac{a}{2}\right)^2 = 12^2 + 5^2 = 144 + 25 = 169 = 13^2$ ✓
• Base area: $a^2 = 10^2 = 100 \text{ cm}^2$
• Lateral surface area: $2al = 2 \times 10 \times 12 = 240 \text{ cm}^2$
• Total surface area: $100 + 240 = 340 \text{ cm}^2$ ✓

Key relationships in square-based pyramid:

• Edge (e): The line from apex to any corner of the base
• Slant height (l): The perpendicular distance from apex to the midpoint of any base side
• Relationship: $e^2 = l^2 + \left(\frac{a}{2}\right)^2$ (Pythagorean theorem)
• Total surface area: $\text{TSA} = a^2 + 2al$ where $a$ is base side and $l$ is slant height

Note: In this problem, we first had to find the slant height using the relationship between the edge, slant height, and half the base side. The edge forms the hypotenuse of a right triangle where one leg is the slant height and the other leg is half the base side.

Question 3

Find the vertical height and the length of the edge of a square based pyramid if its total surface area is 144 cm² and slant height is 5 cm.

Solution:

Given information:

• Total surface area (TSA) = 144 cm²
• Slant height (l) = 5 cm
• Vertical height (h) = ?
• Length of the edge (e) = ?

Step 1: Find the length of base side (a)

We know that total surface area = $a^2 + 2al$

Substituting the given values:

$144 = a^2 + 2a \times 5$ $144 = a^2 + 10a$ $a^2 + 10a - 144 = 0$

Factoring the quadratic equation:

$a^2 + 18a - 8a - 144 = 0$ $a(a + 18) - 8(a + 18) = 0$ $(a + 18)(a - 8) = 0$

This gives us: $a + 18 = 0$ or $a - 8 = 0$

So: $a = -18$ or $a = 8$

Since length cannot be negative, $a = 8$ cm

Therefore, length of side of base (a) = 8 cm

Step 2: Find the vertical height (h)

Using the relationship: $l^2 = h^2 + \left(\frac{a}{2}\right)^2$

Substituting the values:

$5^2 = h^2 + \left(\frac{8}{2}\right)^2$ $25 = h^2 + 16$ $25 - 16 = h^2$ $9 = h^2$ $h = 3$ cm

∴ The vertical height (h) = 3 cm

Step 3: Find the length of edge (e)

Using the relationship: $e^2 = l^2 + \left(\frac{a}{2}\right)^2$

Substituting the values:

$e^2 = 5^2 + \left(\frac{8}{2}\right)^2$ $e^2 = 25 + 16$ $e^2 = 41$ $e = \sqrt{41}$ cm

Final Answer:

Hence, the vertical height (h) = 3 cm and the length of the edge (e) = √41 cm.

Verification:

• Base area: $a^2 = 8^2 = 64 \text{ cm}^2$
• Lateral surface area: $2al = 2 \times 8 \times 5 = 80 \text{ cm}^2$
• Total surface area: $64 + 80 = 144 \text{ cm}^2$ ✓
• Vertical height check: $l^2 = h^2 + \left(\frac{a}{2}\right)^2 = 3^2 + 4^2 = 9 + 16 = 25 = 5^2$ ✓
• Edge length check: $e^2 = l^2 + \left(\frac{a}{2}\right)^2 = 25 + 16 = 41$ ∴ $e = \sqrt{41}$ ✓

Problem-solving approach:

• Step 1: Use total surface area formula to set up quadratic equation for base side
• Step 2: Solve quadratic equation and reject negative solution
• Step 3: Use Pythagorean theorem to find vertical height from slant height
• Step 4: Use Pythagorean theorem to find edge length from slant height

Key geometric relationships:

• Total surface area: $\text{TSA} = a^2 + 2al$ (base + lateral)
• Slant height and vertical height: $l^2 = h^2 + \left(\frac{a}{2}\right)^2$
• Edge and slant height: $e^2 = l^2 + \left(\frac{a}{2}\right)^2$
• Edge and vertical height: $e^2 = h^2 + \left(\frac{a\sqrt{2}}{2}\right)^2$

Note: This is a reverse problem where we work backwards from the total surface area to find the base dimensions, then use geometric relationships to find the vertical height and edge length. The quadratic equation arises naturally from the surface area constraint.

Question 4

The figure given alongside is a square based pyramid. The length of base is 12 cm and the lateral surface area is 240 cm². Find the slant height and vertical height.

Solution:

Given information:

• Length of the base (a) = 12 cm
• Lateral surface area (LSA) = 240 cm²
• Vertical height (h) = ?
• Slant height (l) = ?

Step 1: Find the slant height (l)

We know that lateral surface area = $2al$

Substituting the given values:

$240 = 2 \times 12 \times l$ $240 = 24l$ $l = \frac{240}{24} = 10$ cm

∴ The slant height (l) = 10 cm

Step 2: Find the vertical height (h)

Using the relationship: $l^2 = h^2 + \left(\frac{a}{2}\right)^2$

Substituting the values:

$(10)^2 = h^2 + \left(\frac{12}{2}\right)^2$ $100 = h^2 + 36$ $100 - 36 = h^2$ $64 = h^2$ $h = 8$ cm

∴ The vertical height (h) = 8 cm

Final Answer:

Hence, the slant height (l) = 10 cm and the vertical height (h) = 8 cm.

Verification:

• Lateral surface area check: $2al = 2 \times 12 \times 10 = 240 \text{ cm}^2$ ✓
• Pythagorean relationship: $l^2 = h^2 + \left(\frac{a}{2}\right)^2 = 8^2 + 6^2 = 64 + 36 = 100 = 10^2$ ✓
• Base half-side: $\frac{a}{2} = \frac{12}{2} = 6$ cm
• Right triangle check: $6^2 + 8^2 = 36 + 64 = 100 = 10^2$ ✓

Problem-solving sequence:

• Step 1: Use lateral surface area formula to find slant height directly
• Step 2: Apply Pythagorean theorem with slant height and half-base to find vertical height
• Key insight: Lateral surface area gives us slant height quickly, then geometry gives vertical height

Geometric understanding:

• Lateral surface area: $\text{LSA} = 2al$ where $a$ is base side and $l$ is slant height
• Right triangle formation: Vertical height, half-base, and slant height form a right triangle
• Key relationship: $l^2 = h^2 + \left(\frac{a}{2}\right)^2$ (Pythagorean theorem)
• Alternative verification: Can also check using 6-8-10 Pythagorean triple

Note: This problem demonstrates how lateral surface area directly provides the slant height when the base dimensions are known. The 6-8-10 triangle is a scaled version of the common 3-4-5 Pythagorean triple, which provides a quick verification check.