This image displays six different pyramids, each with dimensions labeled for its base and height. The task is to find the volume of each pyramid.
The formula for the volume $V$ of a pyramid is:
$$ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} $$
Since all the bases shown are triangles, the area of the base can be calculated using the formula for the area of a triangle:
$$ \text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height} $$
Therefore, the volume formula becomes:
$$ V = \frac{1}{3} \times \left(\frac{1}{2} \times \text{base} \times \text{height}\right) \times \text{pyramid height} $$
$$ V = \frac{1}{6} \times \text{base} \times \text{height} \times \text{pyramid height} $$
I will now calculate the volume for each of the six pyramids one by one.
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Pyramid 1 (Top Left):
- Base: 5 cm
- Height of triangular base: 6 cm
- Pyramid height: 5 cm
$$ V_1 = \frac{1}{6} \times 5 \times 6 \times 5 = \frac{150}{6} = 25 \text{ cm}^3 $$
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Pyramid 2 (Top Right):
- Base: 5 cm
- Height of triangular base: 6 cm
- Pyramid height: 15 cm
$$ V_2 = \frac{1}{6} \times 5 \times 6 \times 15 = \frac{450}{6} = 75 \text{ cm}^3 $$
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Pyramid 3 (Middle Left):
- Base: 15 cm
- Height of triangular base: 6 cm
- Pyramid height: 5 cm
$$ V_3 = \frac{1}{6} \times 15 \times 6 \times 5 = \frac{450}{6} = 75 \text{ cm}^3 $$
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Pyramid 4 (Middle Right):
- Base: 2.5 cm
- Height of triangular base: 3 cm
- Pyramid height: 7.5 cm
$$ V_4 = \frac{1}{6} \times 2.5 \times 3 \times 7.5 = \frac{56.25}{6} = 9.375 \text{ cm}^3 $$
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Pyramid 5 (Bottom Left):
- Base: 5 cm
- Height of triangular base: 6 cm
- Pyramid height: 15 cm
$$ V_5 = \frac{1}{6} \times 5 \times 6 \times 15 = \frac{450}{6} = 75 \text{ cm}^3 $$
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Pyramid 6 (Bottom Right):
- Base: 25 cm
- Height of triangular base: 30 cm
- Pyramid height: 75 cm
$$ V_6 = \frac{1}{6} \times 25 \times 30 \times 75 = \frac{56250}{6} = 9375 \text{ cm}^3 $$
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Final Answer:
Here are the volumes for each of the six pyramids:
1.
Top Left: $25 \text{ cm}^3$
2.
Top Right: $75 \text{ cm}^3$
3.
Middle Left: $75 \text{ cm}^3$
4.
Middle Right: $9.375 \text{ cm}^3$
5.
Bottom Left: $75 \text{ cm}^3$
6.
Bottom Right: $9375 \text{ cm}^3$
The solution involves applying the standard volume formula for a pyramid, substituting the given dimensions for each case, and performing the arithmetic calculation.
Parent Tip: Review the logic above to help your child master the concept of volume of pyramids worksheet.
