Flatten two triangles. Find the diagonal.

SparkClass

Rotate a prism, pull out its hidden right triangles, and build the space-diagonal rule one proof at a time.

CCSS 8.G.B.7
Rotatable 3D Construction
Drag or use arrow keys
Rotatable rectangular prism and two extracted right triangles A 3 by 4 by 5 prism. Its base diagonal is 5 and its space diagonal is the square root of 50.
00

Meet the two diagonals. The blue segment e crosses the base; the plum segment d crosses the whole prism.

Explore In 3D

Make the two right angles do the work

The mathematical hinge is not the final formula. It is noticing that the vertical height is perpendicular to the entire base—and therefore to the base diagonal.

1

Predict before calculating

In Explore, ask: Which must be longer, e or d? Can d be longer than l + w + h? Have students defend their predictions from the drawing.

2

Flatten without changing the sides

Play Step 1 slowly. Match the coral, teal, and blue sides before and after extraction. The base right angle sits between l and w, so e² = l² + w².

3

Make the second right angle explicit

In Step 2, the right angle is between e and h—not between d and h. Because h is perpendicular to the base, h is perpendicular to e. Thus d² = e² + h².

4

Substitute the squared quantity

Replace e² with l² + w². There is no need to round e first. For 3 · 4 · 5, d² = 50 and d = √50 = 5√2. Switch to 3 · 4 · 12 for the integer result d = 13.

CCSS 8.G.B.7 alignment: students apply the Pythagorean Theorem to determine an unknown side length in a three-dimensional mathematical problem. The lab derives d² = l² + w² + h² through two right triangles rather than presenting it as a standalone rule.