Sliced
An irregular tetrahedron has two opposite sides the same length a
and the line joining their midpoints is perpendicular to these two
edges and is of length b. What is the volume of the tetrahedron?
Age
14 to 16
Challenge level
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An irregular tetrahedron has two opposite sides the same length ($a$ say) and the line joining their midpoints is perpendicular to these two edges and is of length $b$. What is the volume of the tetrahedron?
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Slice the tetrahedron in two with a cut through the line $b$ and one of the equal edges. The perpendiculars are very important because of what you need to find the volume of a pyramid. You now have two identical pyramids like this:
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Daniel, from Wales High School, sent us this very elegant solution:
The formula for the volume of a tetrahedron is $$1/3 \times \text{area of base} \times \text{perpendicular height}$$ If you slice the tetrahedron in half through $b$ you end up with two equal smaller pyramids. I will work out the volume of one and then multiply by $2$ because they have equal volumes. So to start, I must work out the area of the base. The area of a triangle is calculated as follows: $$1/2 \times \text{base} \times \text{perpendicular height}$$ The base is $a$ and the perpendicular height is $b$ so the area of the base is $$(1/2)\times a\times b=(a b/2)$$ So using the formula for the volume of a pyramid: $$\text{Volume}=(1/3) \times (ab/2)\times (a/2)=a^2b/12$$ Times this by $2$ to get the volume for the big tetrahedron: $$(a^2b/12)\times 2=2a^2b/12 =a^2b/6$$ So the volume of the big tetrahedron is $a^2b/6$.
This is quite a difficult problem to visualise.
A model of the solid made from the two "halves" may help pupls to see the symmetry.