Geometry is the study of shapes, sizes, and the properties of space. This guide covers the essential formulas you'll need for rectangles, triangles, circles, and 3D solids — plus a worked example of the Pythagorean theorem.
A four-sided shape with four right angles. Opposite sides are equal in length.
A three-sided polygon. The interior angles of any triangle always add up to 180°.
A round shape where every point on the edge is the same distance (the radius) from the center.
Area measures the amount of space inside a 2D shape. Here are the key formulas:
| Shape | Formula |
|---|---|
| Rectangle | A = l × w |
| Triangle | A = ½ × b × h |
| Circle | A = π × r² |
| Trapezoid | A = ½ × (a + b) × h |
Perimeter is the total distance around the outside of a shape.
| Shape | Formula |
|---|---|
| Rectangle | P = 2(l + w) |
| Triangle | P = a + b + c |
| Circle (Circumference) | C = 2πr |
Volume measures the amount of space inside a 3D object.
| Shape | Formula |
|---|---|
| Cube | V = s³ |
| Rectangular Prism | V = l × w × h |
| Cylinder | V = π × r² × h |
| Sphere | V = 4/3 × π × r³ |
| Cone | V = 1/3 × π × r² × h |
The Pythagorean theorem applies to right triangles (triangles with one 90° angle). It states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides:
a² + b² = c²
Worked Example:
Find the hypotenuse of a right triangle with sides a = 3 and b = 4.
c² = a² + b²
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5
The hypotenuse is 5.
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