FACEing Geometry Lesson 9 - Free Printable

Let’s solve both problems step by step.

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Problem 19:

We are given a square pyramid.

- Base is a square: 10 mm × 10 mm → so area of base = 10 × 10 = 100 mm²
- Height of the pyramid (the perpendicular height from base to apex) is shown as 12 mm (from the right triangle with legs 5 and 12, hypotenuse 13 — that’s the slant height, but the vertical height is 12 mm).

Volume of a pyramid = (1/3) × base area × height

So:

V = (1/3) × 100 × 12
= (100 × 12) / 3
= 1200 / 3
= 400 mm³

Now check the options:

(a) If V = 400 mm³ → color ears, nose, face apricot. ✔ This matches.

(b) If V = 1200 mm³ → color brown. Not our answer.

→ So for Problem 19, volume is 400 mm³, and we follow instruction (a).

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Problem 20:

Also a square pyramid.

- Base: 12 m × 12 m → base area = 144 m²
- Slant edge (from corner of base to apex) is 8 m? Wait — look at the diagram.

Actually, in the red writing, they’re using Pythagoras on a right triangle where:

One leg is half the diagonal of the base? Or maybe it's simpler.

Wait — the red work shows:

They drew a right triangle inside: one leg is 6 (half of 12), another leg is x (height), hypotenuse is 8? But then they wrote:

6² + x² = 8² → 36 + x² = 64 → x² = 28 → x = √28

That makes sense if the 8 m is the slant height from the center of a side to the apex? No — wait, actually, if you go from the center of the base to a corner, that’s half the diagonal.

Diagonal of base = √(12² + 12²) = √(288) = 12√2 → half-diagonal = 6√2 ≈ 8.485, which is more than 8 — so 8 can’t be the distance from center to corner.

But in the red work, they used 6 and 8 — so likely, the 8 m is the slant height from the midpoint of a base edge to the apex.

In that case, the right triangle has:

- One leg = half the base side = 6 m
- Hypotenuse = slant height = 8 m
- Other leg = height of pyramid = h

Then:

h² + 6² = 8²
h² + 36 = 64
h² = 28
h = √28 = 2√7 m

Now volume:

V = (1/3) × base area × height
= (1/3) × (12 × 12) × √28
= (1/3) × 144 × √28
= 48 × √28

Simplify √28 = √(4×7) = 2√7

So V = 48 × 2√7 = 96√7 m³

Now approximate √7 ≈ 2.6458

So 96 × 2.6458 ≈ let’s compute:

96 × 2.6 = 249.6
96 × 0.0458 ≈ 4.3968
Total ≈ 253.9968 ≈ 254 m³

So volume ≈ 254 m³

Check options:

(a) If V = 762 m³ → outline nose/ears orange, mouth black → not us.

(b) If V = 254 m³ → outline nose/ears black, mouth red → ✔ this matches.

So for Problem 20, volume is approximately 254 m³, follow instruction (b).

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Final Answer:

Problem 19: Volume = 400 mm³ → Follow part (a): color ears, nose, and face apricot.

Problem 20: Volume ≈ 254 m³ → Follow part (b): outline nose and ears in black, and mouth in red.