Let's solve both problems step by step.
---
Surface Area of a Pyramid (with Square Base)
The
surface area of a right pyramid with a square base is:
$$
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
$$
-
Base Area = side² (since the base is a square)
-
Lateral Area = 4 × (Area of one triangular face)
Each triangular face has:
- Base = side of the square base
- Height = slant height (the height of the triangle, not the pyramid’s vertical height)
So,
$$
\text{Lateral Area} = 4 \times \left( \frac{1}{2} \times \text{base} \times \text{slant height} \right) = 2 \times \text{base} \times \text{slant height}
$$
Thus,
$$
\text{Surface Area} = \text{side}^2 + 2 \times \text{side} \times \text{slant height}
$$
---
##
✔ Problem 1:
Given:
- Base side = 8 in
- Slant height = 8.1 in (from the diagram — this is the blue dashed line from apex to midpoint of base edge)
- Vertical height = 7 in (not needed directly since we're given slant height)
We use the formula:
$$
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
$$
Step 1: Base Area
$$
= 8^2 = 64 \text{ in}^2
$$
Step 2: Lateral Area
Each triangular face has:
- Base = 8 in
- Height (slant height) = 8.1 in
Area of one triangle:
$$
= \frac{1}{2} \times 8 \times 8.1 = 4 \times 8.1 = 32.4 \text{ in}^2
$$
Total lateral area (4 triangles):
$$
= 4 \times 32.4 = 129.6 \text{ in}^2
$$
Step 3: Total Surface Area
$$
= 64 + 129.6 = \boxed{193.6} \text{ in}^2
$$
---
##
✔ Problem 2:
Given:
- Base side = 10 mm
- Slant height = 9.4 mm (blue dashed line)
- Vertical height = 8 mm (again, not needed since slant height is given)
Step 1: Base Area
$$
= 10^2 = 100 \text{ mm}^2
$$
Step 2: Lateral Area
One triangular face:
$$
= \frac{1}{2} \times 10 \times 9.4 = 5 \times 9.4 = 47 \text{ mm}^2
$$
Total lateral area:
$$
= 4 \times 47 = 188 \text{ mm}^2
$$
Step 3: Total Surface Area
$$
= 100 + 188 = \boxed{288} \text{ mm}^2
$$
---
✔ Final Answers:
1.
Surface Area = 193.6 in²
2.
Surface Area = 288 mm²
---
🔍 Notes:
- The
vertical height (7 in and 8 mm) is not used here because the
slant height is already provided.
- If only vertical height were given, we'd need to use the Pythagorean Theorem to find slant height:
$$
\text{Slant height} = \sqrt{(\text{vertical height})^2 + \left(\frac{\text{base}}{2}\right)^2}
$$
But here, it's already given, so no need.
Let me know if you'd like to see that method as well!
Parent Tip: Review the logic above to help your child master the concept of area equations worksheet.
