Gym Library
Browse short, clear explanations for every geometry concept in the gym. Read before you train, or look something up when you get stuck.
29 concepts · 55 min total
An angle is formed when two rays meet at a shared endpoint called the vertex. Think of it like two arms stretching out from the same shoulder.
Angles are grouped by their size. Knowing the names makes it faster to describe and reason about shapes.
Two angles are complementary when they add up to exactly 90°. Together they form one right angle.
Two angles are supplementary when they add up to exactly 180°. Together they form a straight line.
When two straight lines cross, they form four angles. The angles directly across from each other are called vertical angles.
Parallel lines never meet. No matter how far you extend them, they stay the same distance apart — like train tracks running off into the horizon.
A polygon is a closed, flat shape with straight sides. "Closed" means the sides connect all the way around — no gaps or open ends.
Every triangle has 3 sides and 3 angles that always sum to 180°. But within that rule, triangles come in several flavours.
A quadrilateral is any polygon with four sides. The four angles always add up to 360°.
The interior angles of any polygon always add up to the same total, and it depends only on the number of sides.
A shape has line symmetry (or reflective symmetry) when you can fold it along a line and both halves match exactly. That fold line is called the axis of symmetry or mirror line.
Perimeter is the distance around the outside of a shape. You measure it by adding up all the side lengths. The result is in regular units — centimetres, metres, inches.
For a rectangle with length l and width w: Perimeter = 2l + 2w. Area = l × w.
The area of a triangle is ½ × base × height.
Every circle has a radius (r) — the distance from the centre to the edge — and a diameter (d = 2r) — the distance all the way across.
Volume measures how much space is inside a 3D object — how much it can hold. Measured in cubic units: cm³, m³.
A rectangular prism (cuboid) has six rectangular faces. Volume = length × width × height. Surface area = 2(lw + lh + wh) — two of each pair of opposite faces.
A cylinder has two circular ends and a curved side. Volume = πr²h — the area of the circular base multiplied by the height.
A net is what a 3D shape looks like when completely unfolded and laid flat. Every face of the shape appears as a flat polygon in the net.
The coordinate plane is a flat grid made by two number lines crossing at right angles. The horizontal line is the x-axis, the vertical line is the y-axis. They cross at the origin: (0, 0).
The x and y axes divide the coordinate plane into four sections called quadrants, numbered I through IV going counter-clockwise from the top right.
A translation slides every point of a shape the same distance in the same direction. The shape doesn't rotate, flip, or resize — it just moves.
A reflection flips a shape over a mirror line. Every point lands on the opposite side of the line, the same distance away.
A rotation turns every point of a shape around a fixed centre point by a given angle. The size and shape stay the same, just the orientation changes.
In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
A Pythagorean triple is a set of three whole numbers that satisfy a² + b² = c². They produce right triangles with no messy decimals.
The Pythagorean theorem can find any missing side of a right triangle, as long as you know the other two.
Two shapes are congruent if they are exactly the same size and shape. One can be moved, rotated, or reflected to fit perfectly on top of the other — no resizing allowed.
Two shapes are similar if they have the same shape but can be different sizes. Their angles are all equal and their sides are proportional.