SME
Grade 8 Math · Problem type
A diagonal through a box
Use this for “Will a 30 cm umbrella fit flat in a 20 cm by 15 cm by 18 cm box, corner to corner?”
Diagram
f² = l² + w², so D² = f² + h²
= l² + w² + h²
D = √(l² + w² + h²)
= √(3² + 4² + 12²) = √169 = 13 cm
f = 5 cm (floor diagonal)
D = 13 cm (long diagonal)
This is an example. Type your own values to start a new problem.
D
2
=
l
2
+
w
2
+
h
2
f
Floor diagonal
worked out (not typed)
Formulas
f
2
=
l
2
+
w
2
(floor
diagonal²
=
length²
+
width²)
5
2
=
3
2
+
4
2
D
2
=
f
2
+
h
2
(long
diagonal²
=
floor
diagonal²
+
height²)
13
2
=
5
2
+
12
2
Assumptions
•
First the diagonal across the floor: its legs are the length and the width.
•
Then the diagonal up through the box: its legs are the floor diagonal and the height.
•
Two right triangles, one after the other. The long diagonal is the longest rod that fits.
Step-by-step
WE KNOW
l = 3 cm (length), w = 4 cm (width), h = 12 cm (height)
FIND
floor diagonal (f), long diagonal (D)
Step 1 · Find floor diagonal (f)
Use f² = l² + w² (floor diagonal² = length² + width²)
Across the floor: the length and width are the legs.
f
2
=
3
2
+
4
2
f
=
√
3
2
+
4
2
f
=
√
9 + 16
f
=
√
25
f
=
5
cm
Step 2 · Find long diagonal (D)
Use D² = f² + h² (long diagonal² = floor diagonal² + height²)
Up through the box: the floor diagonal and the height are the legs.
D
2
=
5
2
+
12
2
D
2
=
25
+
144
=
169
D
=
√
169
D
=
13
cm
Check
Put the numbers back in. Both sides should match.
5
2
=
3
2
+
4
2
✓
13
2
=
5
2
+
12
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Nothing to review first. This is a starting point.
Related lessons
Pythagorean theorem and distance between points
Main lesson
Distance between two points
Use this for “Find the distance between (−9, −12) and (0, 0)” or “How long is the segment on the grid?”
Is it a right triangle?
Use this for “Which of these could be the sides of a right triangle?” with the converse of the Pythagorean theorem.