Circles and Arcs

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The content aids in vocabulary acquisition for both English and Japanese learners.
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Through the content, learners grasp words and phrases more intuitively because they already know their meanings in their native language.
It is like learning the word “apple” by looking at a picture of one.
STEM-Plus Content facilitates easy learning of abstract yet useful words.

To ensure effective and efficient learning, the content is designed to be concise and illustrative.
Learners can easily understand the material through images and examples.

In addition, the content provides useful tips on math and other STEM subjects, supporting
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How to use:
Carefully selected English words and phrases related to “Circles and Arcs” are highlighted in yellow in the images below. The words are also summarized at the bottom of this page.
We recommend reviewing them regularly during your daily math studies to enhance your vocabulary.

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  • A circle is the set of all points that are equidistant from a given point called the center of the circle.
  • A radius is a segment form the center of the circle to any point of the circle.
  • A chord is a segment whose endpoints are points on the circle.
  • A diameter is a chord that passes through the center of the circle.
  • The terms radius and diameter describe measures as well as segments.
Definition of circle, center, radius, chord, and diameter


Circles and Lines (Tangents, etc.)

  • A secant is a line that intersects a circle in two points.
  • A tangent is a line in the plane of a circle that intersects the circle in exactly one point, the point of tangency.
Definition of tangent

Circles and Tangents

Circles and Circles (Internal Tangency, etc.)

  • Coplanar circles that intersect in one point are called tangent circles.
  • Coplanar circles that have a common center are called concentric circles.
  • Internally tangent / Externally tangent
    接する / 接する
Definition of concentric circles and tangent

Coplanar Circles

Arcs and Measurement

  • An arc of a circle is two points on the circle and the continuous part of the circle between the two points.
  • A central angle of a circle is an angle whose vertex is the center of the circle.
  • Measuring Arcs

    • The measure of an arc is defined to be the measure of its central angle.
    • The expression m(AB) is read as “the measure of arc AB.”
    • The measure of the entire circle is 360°. The measure of a semicircle is 180°.
      全体での大きさは360°。 半円での大きさは180°。
Definition of arc and its measure

Measuring Arcs

Circumference and Arc Length

  • The distance around a circle is called the circumference.
  • For every circle, the ratio of the circumference to the diameter is close to 3.14.
    The Greek letter π (pi) is used to represent this ratio.
  • The circumference C of a circle is the product of π and the diameter d, or twice the product of π and the radius r.
  • An arc length is a portion of the circumference of a circle.
  • The ratio of the length of an arc to the circumference is equal to the ratio of the measure of the arc to 360°.
Definition of circumference and arc length.

Circumference and Arc Length

Area of Circles and Sectors

  • The area A of a circle is the product of π and the square of the radius r.
  • A sector of a circle is the region bounded by two radii of the circle and their intercepted arc.
    おうぎ形とはふたつの半径とそれらの間の弧を境界とする 領域のこと。
  • The ratio of the area of a sector of a circle to the area of the whole circle is equal to the ratio of the measure of the intercepted arc to 360°.
Explaining how to find the area of a circle and a sector

Areas of Circles and Sectors


diameter 直径

describe を言い表す

measure 大きさ;測定単位;対策/を測る、長さなどを見つける;を評価する

intersection 交差点;交差
📎intersect (と)交差する/交差

concentrate 集中する{注意など};集結する{場所}
📎concentric 中心が同じ

internal 内部の;国内の;体内の

external 外部の;外国の;外的な

common 共通の;普通の

define を定義する;を特徴づける

express を表現する
📎expression 表現

entire 完全な、全部の

distance 距離{2点の間の空間の大きさ}

length 長さ{物の両端の間や時間の}

portion 部分;{食事の}一人前;{責任などの}割当て

region 領域;地域

boundary 境界
📎bound {be bounded by A}Aを境界とする;跳ねる;bind(縛る)の過去・分詞



Making the most of ‘Tangent’

The term ‘tangent’ is used to refer to the trigonometric function ‘tan’ (tangent) as well as to mean a line that intersects a circle in exactly one point (‘tangent line’). Interestingly, the ‘tan’ function actually relates to the length of a tangent line. What am I talking about? Look forward to high school mathematics.

By the way, the etymological meaning of ‘tangent’ is ‘to touch,’ and the word ‘tangible’ comes from the same root. The term ‘tangible’ is frequently used in various professional contexts. It’s a good idea to learn these terms simultaneously, making the most of this opportunity.

tangible {目に見えて}明らかな;触れられる
tangible assets 有形資産、intangible assets 無形資産

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