1To measure, gauge, or weigh — to determine a quantity (time, length, weight, etc.) using instruments or a fixed standard. The kanji vary by what's being measured: 計る for time and numerical/calculated quantities, 測る for distance, height, depth, temperature, etc., 量る for weight and volume. In speech the distinction is often blurred, but standard writing chooses the kanji to fit the object.
ストップウォッチで走るタイムを計った。
I timed my run with a stopwatch.
子供の身長を測ったら、去年より3センチ伸びていた。
When I measured my child's height, they had grown three centimeters since last year.
毎朝、体温を測ってから出勤する。
I take my temperature every morning before heading to work.
パンの材料はキッチンスケールで正確に量ろう。
Let's weigh the bread ingredients accurately with a kitchen scale.
プールの深さを測ってから子供を遊ばせた。
I let the kids play after I had checked how deep the pool was.
2To gauge, surmise, or conjecture — to estimate something intangible such as another person's feelings, intentions, or the prospects of a situation. Appears in expressions like 推し量る ("to surmise"), 真意を計る ("to read someone's true intent"). Somewhat formal/written in tone.
彼の表情から本心を計るのは難しかった。
It was difficult to read his true feelings from his expression.
相手の気持ちを推し量って、慎重に言葉を選んだ。
Mindful of the other person's feelings, I chose my words carefully.
この計画が成功するかどうかは、まだ計りかねる。
Whether this plan will succeed is still hard to gauge.