Surface Area & Volume of Square Pyramids worksheet with a diagram and QR code for self-checking.

Let’s solve this step by step.

We are given a square pyramid with:

- Base side length = 14 inches
- Height of the pyramid (from base to apex, perpendicular) = 24 inches
- The slant height is NOT given directly — but we can find it using the Pythagorean theorem if needed. However, looking at the diagram and labels:

Wait — in the diagram, the “24 in” is labeled as the height from the center of the base to the apex, which is the vertical height (H).

The base is square, so each side is 14 in → that means the base perimeter P = 4 × 14 = 56 in

The base area B = 14 × 14 = 196 in²

Now, for lateral surface area, we need the slant height (ℓ) — that’s the height of one triangular face, from the midpoint of a base edge up to the apex.

To find ℓ, imagine a right triangle inside the pyramid:

- One leg = vertical height H = 24 in
- Other leg = half the base side = 14 ÷ 2 = 7 in
- Hypotenuse = slant height ℓ

So use Pythagoras:

ℓ² = 24² + 7²
ℓ² = 576 + 49 = 625
ℓ = √625 = 25 in ✔

Great! Now we have all values:

P = 56 in
ℓ = 25 in
H = 24 in
B = 196 in²

---

Part 1: Lateral Area & Surface Area

Lateral Area (LA) = (1/2) × Perimeter × Slant Height
= (1/2) × 56 × 25
= 28 × 25
= 700 in²

Surface Area (SA) = Lateral Area + Base Area
= 700 + 196
= 896 in²

---

Part 2: Volume

Volume of pyramid = (1/3) × Base Area × Height
= (1/3) × 196 × 24

First, 196 × 24:

Break it down:
196 × 20 = 3920
196 × 4 = 784
Total = 3920 + 784 = 4704

Then divide by 3:
4704 ÷ 3 = 1568 in³

✔ Check: 3 × 1568 = 4704 → correct.

---

Final Answer:

P = 56
ℓ = 25
H = 24
B = 196

1. Lateral Area = 700 in², Surface Area = 896 in²
2. Volume = 1568 in³