Geometry Formulas and Concepts
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.
Basic Shapes
Rectangle
A four-sided shape with four right angles. Opposite sides are equal in length.
Triangle
A three-sided polygon. The interior angles of any triangle always add up to 180°.
Circle
A round shape where every point on the edge is the same distance (the radius) from the center.
Area Formulas
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 Formulas
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 Formulas
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
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.
Put these formulas to work
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