It looks like you've uploaded an image of a
square pyramid with some dimensions labeled. Let's analyze the problem and solve it step by step.
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🔍 Understanding the Figure
The figure shows a
square pyramid, meaning:
- The base is a
square.
- The apex (top point) is connected to all four corners of the square base.
- One edge of the base is labeled as
5 cm, so each side of the square base is 5 cm.
- A slant height (from the apex to the midpoint of one of the base edges) is labeled as
8 cm.
We are likely being asked to find either:
- The
volume of the pyramid, or
- The
surface area, or
- Possibly the
height of the pyramid.
Since no specific question was stated, I'll assume the most common task:
find the volume of the pyramid.
---
✔ Step 1: Volume of a Pyramid
The formula for the volume $ V $ of a pyramid is:
$$
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
$$
We know:
- Base is a square with side 5 cm → Base Area = $ 5 \times 5 = 25 \, \text{cm}^2 $
- But we don’t yet know the
vertical height (the perpendicular distance from the apex to the base).
We’re given the
slant height = 8 cm.
This slant height is the distance from the apex to the
midpoint of a base edge, forming a right triangle with:
- The vertical height $ h $ (unknown),
- Half the base edge = $ \frac{5}{2} = 2.5 \, \text{cm} $,
- And the slant height = 8 cm (hypotenuse).
---
✔ Step 2: Use Pythagoras’ Theorem
In the right triangle formed by:
- Vertical height $ h $
- Half base = 2.5 cm
- Slant height = 8 cm
$$
h^2 + (2.5)^2 = 8^2
$$
$$
h^2 + 6.25 = 64
$$
$$
h^2 = 64 - 6.25 = 57.75
$$
$$
h = \sqrt{57.75} \approx 7.6 \, \text{cm}
$$
---
✔ Step 3: Compute Volume
$$
V = \frac{1}{3} \times 25 \times \sqrt{57.75}
$$
Let’s compute it numerically:
$$
\sqrt{57.75} \approx 7.6
$$
$$
V \approx \frac{1}{3} \times 25 \times 7.6 = \frac{190}{3} \approx 63.33 \, \text{cm}^3
$$
So, the
volume is approximately 63.3 cm³.
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🧮 Optional: Exact Value
$$
h = \sqrt{57.75} = \sqrt{\frac{231}{4}} = \frac{\sqrt{231}}{2}
$$
$$
V = \frac{1}{3} \times 25 \times \frac{\sqrt{231}}{2} = \frac{25\sqrt{231}}{6} \, \text{cm}^3
$$
Approximately:
$$
\sqrt{231} \approx 15.198 \Rightarrow V \approx \frac{25 \times 15.198}{6} \approx \frac{379.95}{6} \approx 63.33 \, \text{cm}^3
$$
---
✔ Final Answer:
>
Volume ≈ 63.3 cm³
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📌 Summary
-
Base side: 5 cm → Base area = 25 cm²
-
Slant height: 8 cm
-
Vertical height: $ \sqrt{8^2 - 2.5^2} = \sqrt{57.75} \approx 7.6 $ cm
-
Volume: $ \frac{1}{3} \times 25 \times 7.6 \approx 63.3 \, \text{cm}^3 $
Let me know if you wanted surface area instead!
Parent Tip: Review the logic above to help your child master the concept of surface area pyramids worksheet.
