eMathematician.com

Circle: arc, sector and segment

Mon, 30 Aug 2021 05:18:00 +0000

A circle has these main slices: arc, sector and segment. In this article we will show how to calculate these parts.

The arc

The arc is any part of a circle. It can be the part of section or segment. The arc length can be calculated by this formula below:

$$L=frac{pi R}{180}cdot theta$$ The arc of a circle.

The sector

The sector of a circle is region bounded by two radius of equal length with a common center. The sector area can be calculated by this formula below:

$$A=frac{pi R^2}{360}cdot theta$$ The sector of a circle.

The segment

The segment of a circle is a region bounded by a chord and one of the arcs connecting the chords endpoints. The area of a segment is the area of a sector minus the triangular. The segment area can be calculated by this formula below:

$$A=frac{pi R^2}{360}cdot theta-frac{R^2 sintheta}{2}$$ The segment of a circle.

Circle: radius, diameter, circumference and area

Tue, 24 Aug 2021 20:40:18 +0000

A circle is a shape consisting of all points are the same distance from the center.

A circle with radius and diameter.

The radius is the distance between any point of the circle and the centre. The diameter is a line segment which goes straight across the circle and passes through the centre. The diameter is twice the radius: $D=2R$. The circumference is the distance around the circle. It can be calculated by these formulas below:

$$C=pi D=2pi R$$

The area enclosed by a circle can be calculated by this formula below:

$$A=pi R^2$$

Solved examples

What is the circumference of a circle with radius of 3.5 cm? Round your answer to the nearest one.

$$C=2pi R=2cdot 3.14cdot 3.5 approx 22 text{ cm}$$

What is the area of a circle with diameter of 5 cm? Round your answer to the nearest one.

$$A=pi R^2=3.14cdot left (frac{5}{2} right )^2 approx 20 text{ cm}^2$$

What is the area of a circle with circumference of 30 cm? Round your answer to the nearest one.

$$C=2pi R, 30=2pi R, R=frac{15}{pi}$$ $$A=pi R^2=3.14cdot left (frac{15}{3.14} right )^2 approx 72 text{ cm}^2$$

The law of cosines and examples

Mon, 16 Aug 2021 14:01:09 +0000

The law of cosines is very useful for solving various tasks with not right triangles.

The law of cosines works for any triangle. The only important thing is that side a faces angle A, side b faces angle B and side c faces angle C. It helps for correct using of formulas.

$$text{c}^2=text{a}^2+text{b}^2-2text{a}text{b}cos{text{C}}$$ Triangle. A, B, C are angles and a, b, c are sides.

The formula above can be rewritten into these formulas, if we need to find other sides of triangle.

$$text{a}^2=text{b}^2+text{c}^2-2text{b}text{c}cos{text{A}}$$ $$text{b}^2=text{a}^2+text{c}^2-2text{a}text{c}cos{text{B}}$$

It is not very necessary but for easier finding of triangle angles we can use these formulas below:

$$cos{text{C}}=frac{text{a}^2+text{b}^2-text{c}^2}{2text{a}text{b}}$$ $$cos{text{B}}=frac{text{a}^2+text{c}^2-text{b}^2}{2text{a}text{c}}$$ $$cos{text{A}}=frac{text{b}^2+text{c}^2-text{a}^2}{2text{b}text{c}}$$

Solved examples

For triangle below find the side a. Round your answer to the nearest tenth.

$$text{a}^2=text{b}^2+text{c}^2-2text{b}text{c}cos{text{A}}$$ $$text{a}^2=text{17}^2+text{21}^2-2text{17}cdottext{21}cdotcos{text{55°}}$$ $$aapprox 17.9$$

For triangle below find the angle A. Round your answer to the nearest one.

$$cos{text{A}}=frac{text{b}^2+text{c}^2-text{a}^2}{2text{b}text{c}}$$ $$cos{text{A}}=frac{text{29}^2+text{39}^2-text{35}^2}{2cdottext{29}cdottext{39}}=0.5026...$$ $$angle text{A}=cos^{-1}(0.5026...)approx 60°$$

The law of sines and examples

Fri, 13 Aug 2021 12:48:32 +0000

The law of sines is very useful for solving various tasks with not right triangles.

The law of sines works for any triangle. The only important thing is that side a faces angle A, side b faces angle B and side c faces angle C. It helps for correct using of formulas.

$$frac{text{a}}{sintext{A}}=frac{text{b}}{sintext{B}}=frac{text{c}}{sintext{C}}$$ Triangle. A, B, C are angles and a, b, c are sides.

Solved examples

For triangle below find the side a. Round your answer to the nearest tenth.

$$frac{text{a}}{sintext{45°}}=require{cancel} cancel{frac{text{b}}{sintext{B}}}=frac{text{40}}{sintext{35°}}$$ $$a=frac{text{40}}{sintext{35°}}cdot sintext{45°} approx 39.2$$

For triangle below find the angle A. Round your answer to the nearest one.

$$frac{text{18}}{sintext{A}}=frac{text{23}}{sintext{75°}}=require{cancel} cancel{frac{text{c}}{sintext{C}}}$$ $$sintext{A}=frac{18cdot sintext{75°}}{23}= 0.7559...$$ $$angle text{A}=sin^{-1}(0.7559...)approx 49°$$

Rewriting radicals into exponential form

Mon, 02 Aug 2021 21:08:27 +0000

We can rewrite every radical as an exponent. Frequently this step helps easily to solve the task of complex radicals.

By using the following property we can rewrite radicals into exponential form and on the contrary.

$$a^{frac{m}{n}}=sqrt[n]{a^m}=(sqrt[n]{a})^m$$

Solved examples

Rewrite radical $sqrt[5]{x^3}$ into exponential form.

$$sqrt[5]{x^3}=x^{frac{3}{5}}$$

Simplify the expression $sqrt{x}cdot sqrt[3]{x}: sqrt[5]{x^2}$

$$sqrt{x}cdot sqrt[3]{x}: sqrt[5]{x^2}=x^{frac{1}{2}}cdot x^{frac{1}{3}}: x^{frac{2}{5}}=x^{frac{1}{2}+frac{1}{3}-frac{2}{5}}=x^{frac{13}{30}}=sqrt[30]{x^{13}}$$

Simplify the expression $sqrt{xsqrt[3]{x^2}}$

$$sqrt{xsqrt[3]{x^2}}=x^{frac{1}{2}}cdot (x^{frac{2}{3}})^{frac{1}{2}}=x^{frac{1}{2}+frac{2}{3}cdot frac{1}{2}}=x^{frac{5}{6}}=sqrt[6]{x^{5}}$$