Grade 8 · Grade 8: Transformations, Pythagoras and Volume · Transformations, Pythagoras and Volume
The Pythagorean Theorem and Curved Solids
Prove and apply a² + b² = c² in two and three dimensions, then measure cones, cylinders and spheres.
Learning objectives
- Explain a proof of the Pythagorean theorem and its converse.
- Find unknown sides in 2D and 3D and distances in the coordinate plane.
- Use the volume formulas for cones, cylinders and spheres.
AERO Mathematics alignment
AERO.M8.GM.5
Explain a proof of the Pythagorean theorem and its converse.
AERO.M8.GM.6
Apply the Pythagorean theorem to determine unknown side lengths in right triangles in two and three dimensions, and to find the distance between two points in the coordinate plane.
AERO.M8.GM.7
Know and use the formulas for the volumes of cones, cylinders and spheres to solve real-world and mathematical problems.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
A builder wants to check that a corner is exactly square. She measures 3 m along one wall, 4 m along the other, and then the diagonal between the marks.
Think about it
What diagonal length proves the corner is square, and why?
Hint: Squares built on the three sides are the key.
EXPLAIN
a² + b² = c² and curved solids
In any right triangle the square on the hypotenuse equals the sum of the squares on the other two sides: a² + b² = c². For the builder, 3² + 4² = 9 + 16 = 25 = 5², so a diagonal of exactly 5 m proves the corner is square.
A visual proof
Four identical right triangles arranged inside a square of side (a + b) leave an inner square of side c. Rearranging the same four triangles instead leaves two squares of side a and b. Same total area, so a² + b² = c².
The converse
If a² + b² = c² for the three side lengths, the triangle must be right-angled. This is what makes the 3-4-5 check work on a building site.
Distance in the coordinate plane
The distance between (x₁, y₁) and (x₂, y₂) is √((x₂ − x₁)² + (y₂ − y₁)²) — the Pythagorean theorem applied to a right triangle drawn on the grid.
Volumes of curved solids
| Solid | Volume | Note |
|---|---|---|
| Cylinder | V = πr²h | Base area times height |
| Cone | V = ⅓πr²h | One third of its cylinder |
| Sphere | V = (4/3)πr³ | Depends on r cubed |
Common misconception
a² + b² = c² only applies to right triangles, and c must always be the hypotenuse — the side opposite the right angle.
INVESTIGATION
Try it yourself
Use the 3-4-5 method to test a real corner in your home. Then find the diagonal of a room in three dimensions (floor diagonal, then height) and check your prediction with a tape measure.
Watch
The Pythagorean Theorem
Math Antics · 10:00
A visual proof that makes a² + b² = c² memorable.
Before you watch: Which side is always the hypotenuse?
- Find the hypotenuse when legs are 9 and 12.
- Find the distance between (1, 2) and (7, 10).
Interactive simulation · PhET
Area Builder
Build squares on the three sides of a right triangle and compare their areas.
While you explore
- How do the square areas demonstrate a² + b² = c²?
- Does the relationship still hold for a non-right triangle? Test it.
Key vocabulary
- Distance formula
- A way to find how far apart two points are on a grid.
- Hypotenuse
- The longest side of a right triangle.
Practice questions
0/1 correct
Level 1 · Criterion A
A right triangle has legs 9 and 12. What is the hypotenuse?
MYP criterion tasks
Level 2 · Criterion D
A cylinder has radius 5 cm and height 12 cm. Find its volume, then find the volume of a cone with the same base and height.
Reflect & track
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