Middle school maths, year 1 · unit 7
Plane and Solid Figures
This unit has a lot of formulas. But before memorising one, it is enough to see it once. A pentagon is three triangles, a slice of pizza is some fraction of the whole, and a cone is a third of a cylinder. We start by splitting up polygons.
How many triangles make a polygon?
Joining boardDraw diagonals from one corner and a polygon splits into triangles. Drag the gold handle to another corner to draw a line.
What fraction of the pizza is one slice?
A board to adjustCut a bigger slice and its edge gets longer and its area larger. By how much? Drag the gold handle to change the angle of the slice.
Spin it and it becomes a solid
A board to arrangeA revolving door, spinning, looks like a cylinder. Drag a paper piece to the axis and spin it.
A pyramid is a third of a prism
A board to arrangeHere are a pyramid and a prism with the same base and height. Fill the pyramid with water and pour it into the prism: how many times to fill it? Drag the scoop over the tank.
Try it yourself
Use what you found on the boards. The numbers change every time and the easy questions come first. Get three right and the lamp lights.
What to take away from this unit
- An n-sided polygon is split by (n − 3) diagonals from one corner into (n − 2) triangles. So its interior angles add up to 180° × (n − 2). The exterior angles always add up to 360°.
- Circumference ÷ diameter is pi, π. A circle of radius r has circumference 2πr and area πr².
- The arc length and area of a sector are directly proportional to its central angle. Work out what fraction of 360° the angle is, then multiply the circumference and the area by it.
- Turn a flat shape once about an axis and you get a solid of revolution: a rectangle gives a cylinder, a right-angled triangle a cone, a semicircle a sphere. Solids bounded only by flat faces are polyhedra.
- The volume of a prism or cylinder is base area × height, and a pyramid or cone holds 1/3 of that. A sphere of radius r has volume 4/3 πr³ and surface area 4πr².