promise, approximately, shrink
約 centers on binding or tightening something: a promise binds people, an approximate number is a tightened or simplified figure, and shrinking makes something tighter or smaller.
約 combines 糸 (thread) with 勺, which likely contributes the sound and the idea of binding or wrapping. The original sense of tying with thread extends to abstract binding like promises and to making things compact.
Thread 糸 wrapped around a ladle 勺: imagine binding a ladle tightly with thread to keep it from moving. That binding is like a promise, and the tight wrapping shrinks the space around it.
For ヤク, picture a yak making a promise by tying a thread around its horn: yak -> ヤク, and the thread binds the promise.
approximately; about
contract; compact; agreement
reservation; appointment; booking; advance order
promise; agreement; arrangement; one's word; contract; pact
treaty; pact; convention; agreement
engagement; betrothal
economising; saving
agreement; rules; code; protocol; convention; bylaw
limitation; restriction; condition; constraints
cancellation of a contract
putting together (e.g. opinions); gathering; collecting; aggregating; summarizing; integrating
summary; digest
public commitment; public promise; campaign pledge; official vow
oath; vow; pledge; covenant
firm promise; definite promise
old promise; old agreement; old covenant
special contract; special agreement; rider (insurance)
secret agreement
thrift; economy; frugality
oath; pledge; pact; covenant; alliance
agreement; stipulation; article; clause
conclusion of a treaty
conclusion of a contract
pre-order; advance order