Drawing Triangles from Given Conditions, and Slicing Solids

Three measurements, three possible outcomes

Give someone three measurements and ask them to draw the triangle. Three things can happen.

Outcome	Meaning
one triangle	every correct drawing is congruent to every other
many triangles	the conditions leave something free to vary
no triangle	the measurements contradict each other

Which outcome you get depends on which three measurements were given, not on how carefully anyone draws.

The sets that fix exactly one triangle

Given	Name	Result
three sides	SSS	one triangle
two sides and the angle between them	SAS	one triangle
two angles and any side	ASA or AAS	one triangle

Take SSS. Draw the longest side. Swing an arc of the second length from one end and an arc of the third from the other. The arcs meet at one point above the line, and that point is the third corner.

Nothing was chosen along the way, which is why every correct drawing comes out congruent.

SAS is the same story: the angle fixes the direction of the second side, and its length fixes where it stops.

ASA fixes both directions from a known side, and two straight lines that are not parallel cross exactly once.

Why some sets leave many triangles

Three angles (AAA) fix the shape and leave the size open.

$$\text{angles } 60°, 60°, 60° \;\Rightarrow\; \text{every equilateral triangle}$$

A side of $1$ cm and a side of $100$ cm both work. All the drawings are similar but only one size can be congruent to yours, so the conditions do not fix a triangle.

Two sides and an angle that is not between them (SSA) can allow two different triangles. Swinging the arc for the third vertex can cross the far line in two places, giving one acute triangle and one obtuse.

That is why SAS names the angle between the sides. Moving the angle elsewhere loses the guarantee.

When no triangle fits

Two conditions rule a triangle out before you start.

The sides cannot reach. The two shorter sides must add to more than the longest, or the arcs never meet.

$$2 + 3 < 6 \;\Rightarrow\; \text{no triangle with sides } 2, 3, 6$$

Picture the two short sides swinging down onto the long one. Even laid flat they span only $5$, leaving a gap of $1$.

The angles overshoot. The three angles of a triangle total $180°$.

$$90° + 60° + 50° = 200° \;\Rightarrow\; \text{no such triangle}$$

Checking these two before drawing saves the drawing. A set that fails either test cannot be rescued by a steadier hand.

Slicing a solid

Cut a solid with a flat plane and look at the cut face. That flat shape is a cross-section, and it depends on where the plane goes.

A right rectangular prism

Slice	Cross-section
parallel to a face	a rectangle identical to that face
perpendicular to the base, straight across	a rectangle
slanted through four faces	a rectangle wider than the box
through three faces near a corner	a triangle

Every slice parallel to a face repeats that face exactly. This is what makes a prism a prism: the same cross-section all the way along, which is also why its volume is base area times height.

A right rectangular pyramid

Slice	Cross-section
parallel to the base	a smaller rectangle, same proportions
through the apex, down to the base	a triangle
slanted across the sides	a four-sided figure, not a rectangle

A pyramid narrows toward its apex, so parallel slices shrink as they rise. That is the difference from a prism, and it is the reason a pyramid’s volume is a third of the prism around it rather than the whole of it.

Worked examples

Common mistakes

Practice problems