Basic Geometry and Formulas

Learning Objectives

Calculate the perimeter and area of common 2D shapes (squares, rectangles, circles).

Calculate the volume of common 3D shapes (cubes, rectangular prisms, spheres).

Identify key terms like radius, diameter, and pi.

Describing Shapes and Space

Geometry is the branch of mathematics that deals with shapes, sizes, positions of figures, and the properties of space. In science, we often need to measure the dimensions of objects, from microscopic cells to planets.

2D Shapes: Perimeter and Area

Perimeter is the total distance around the outside of a two-dimensional shape. You find it by adding up the lengths of all the sides.

Area is the amount of surface a two-dimensional shape covers.

Square (side length s)

Perimeter = 4s

Area = s²

Rectangle (length l and width w)

Perimeter = 2l + 2w

Area = l * w

Circle

Radius (r): The distance from the center to any point on the circle.

Diameter (d): The distance across the circle through the center. (d = 2r)

Pi (π): A special irrational number, approximately 3.14159. It is the ratio of a circle's circumference to its diameter.

Circumference (C): The perimeter of a circle.

C = πd or C = 2πr

Area (A):

A = πr²

3D Shapes: Volume

Volume is the amount of space a three-dimensional object occupies.

Cube (side length s)

Volume = s³

Rectangular Prism (length l, width w, and height h)

Volume = l w h

Sphere (radius r)

Volume = (4/3)πr³

Cylinder (radius r and height h)

Volume = (Area of base) height = (πr²) h

Example Calculation:

Find the volume of a sphere with a radius of 3 cm.

Formula: V = (4/3)πr³

Substitute: V = (4/3) π (3 cm)³

Calculate: V = (4/3) π (27 cm³)

Simplify: V = 36π cm³

Approximate: V ≈ 36 * 3.14 ≈ 113.04 cm³

Key Terms