Geometry
Proofs you can see: triangles, circles, similarity, area and volume.
Six theorems, each proved from a picture. The angle sum of a triangle comes first because everything else uses it. Pythagoras and similar triangles give the tools for lengths, and the circle and the solids at the end need an argument about limits that prepares the ground for calculus.
Start with lesson 1-
1
Why the angles of a triangle add to 180°Tear the corners off any triangle and they line up along a straight edge. A single parallel line explains why, and a sphere shows the line is needed.
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2
The Pythagorean theorem: a proof you can seeWhy the squares on the two short sides of a right triangle add up to the square on the long side, proved by moving four triangles inside a box.
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3
Why the area of a circle is πr²The number π is defined by the distance around a circle. Cutting the circle into sectors, and then into rings, shows why the same π gives its area.
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4
Similar triangles and why area grows as the squareScaling multiplies every length by one factor and keeps every angle. That explains shadows, maps, and why a pizza twice as wide is four times the pizza.
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5
The inscribed angle theorem and Thales' semicircleFrom every point on one arc of a circle, a chord is seen under the same angle, half the angle at the center. Two isosceles triangles explain why.
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6
Why a cone is a third of a cylinder, and a sphere's volumeA cone holds exactly one third of the cylinder around it. A cube cut into three pyramids shows why, and the same slicing idea gives the volume of a sphere.
