Category Archives: Heron’s Law

December 10, 2025 · 7:54 am

Geometry Problem

Problem from 50 Maths Problems with Solutions – Ajmal KP

The blue shaded area is the area of triangles APO and AQO subtract the sector POQ.

We can use Heron’s law to find the area of the triangle \Delta{ABC}

    \begin{equation*}A=\sqrt{s(s-a)(s-b)(s-c)}\end{equation}

where s=\frac{a+b+c}{2}

    \begin{equation*}A=\sqrt{20(20-16)(20-10)(20-14)}=40\sqrt{3}\end{equation}

We also know the area of triangle \Delta{ABC}=sr where r is the radius of the inscribed circle.

Hence, 40\sqrt{3}=20r and r=2\sqrt{3}

We know AP=AQ, CQ=CR, and BP=BR – tangents to a circle are congruent.

    \begin{equation*}14-x=6+x\end{equation}

(1)   \begin{equation*}8=2x\end{equation*}

(2)   \begin{equation*}x=4\end{equation*}

Area \Delta{AQO}=\frac{1}{2}10\times 2\sqrt{3}=10\sqrt{3}

Area \Delta{APO}=Area \Delta{AQO}

    \begin{equation*}tan(\theta)=\frac{10}{2\sqrt{3}}\end{equation}

    \begin{equation*}\theta=70.9^{\circ}\end{equation}

Area of sector OPQ=\frac{2\times70.9}{360}\pi (2\sqrt{3})^2=14.8

Blue area = 20\sqrt{3}-14.8=19.8cm^2

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Filed under Algebra, Area, Finding an angle, Finding an area, Geometry, Heron's Law, Interesting Mathematics, Puzzles, Radius and Semi-Perimeter, Right Trigonometry, Solving Equations, Trigonometry

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March 29, 2024 · 6:20 am

Three Circles – Area Problem

This is question 5 from the UK Maths Trust Senior Challenge October 2023.

I have tackled this in three ways; using non-right trig to find the area, Heron’s Law, and the Shoelace Formula.

Method 1

Use the area of a triangle formula

Use the cosine rule to find cosθ.

Once we have cosθ, we can find sinθ.

Hence the area is,

Method 2

Use Heron’s law.

Heron’s law is a way of calculating area of a triangle from the lengths of the three sides of the triangle.

This is my preferred method – simple and direct.

Method 3

Shoelace formula (Gauss’s Area formula)

We need to allocate each of the vertices a co-ordinate.

The co-ordinates are listed in an anti-clockwise direction.

This is probably a bit over the top, but once you get the hang of it, it’s very easy.

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Filed under Area, Area of Triangles (Sine), Heron's Law, Non-Right Trigonometry, Shoelace Forumla, Trigonometry

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