Circle theorems

Inscribed angles, tangents, chords, and the angle in a semicircle.

An angle inscribed in a circle is half the central angle on the same arc; an angle in a semicircle is a right angle; a tangent meets the radius at 90°; equal chords are equidistant from the centre. Picture it: slide the inscribed angle's vertex around the arc and its size never changes. Think it: all of these are one fact, the central angle theorem, applied to different configurations.

Àwọn ààtò àwọn ìṣàmúlò-ètò · circle radius 5

Geometry · Circles

Circle radius 5

Àwọn ìṣàmúlò-ètò

r = 5
Everything about a circle follows from its radius.
d = 2r = 10
Diameter.
C = 2\pi r = 10 \pi \approx 31.416
Circumference.
A = \pi r^2 = 25 \pi \approx 78.540
Area.

Fi àwọn àgbèwọlé hàn

C = 10 \pi,\quad A = 25 \pi

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Bii o ṣe le: Circle theorems

Find the centre and draw radii to every point on the circle in the problem.
Look for isosceles triangles (two radii). Their base angles are equal.
Apply the inscribed-angle or tangent, radius theorem to relate the angles, then solve.

Àwọn Àtòjọ-ẹ̀yàn

Why does every triangle have angles adding to 180°?

Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.

When do I use the law of sines versus the law of cosines?

Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.

What is the difference between area and perimeter?

Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.

Diẹ̀ nínú Geometry

Triangles Pythagorean theorem Circles Polygons Solids Coordinate geometry Euclid's axioms and the structure of proof Congruent and similar triangles Transformations: translation, rotation, reflection, dilation Solid geometry: prisms, pyramids, spheres