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Home » Area of Geometric Shapes » Equilateral Triangle: Find its Area from the Radius of an Inscribed Circle

Equilateral Triangle: Find its Area from the Radius of an Inscribed Circle

Last updated: Sep 30, 2019 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

This problem is the opposite of the one where we found the area of the inscribed circle using the length of the side of the equilateral triangle. Here we will use the circle's radius to find the area of the triangle.

Problem

Circle O with radius r is inscribed in an equilateral triangle. Find the area of the triangle in terms of r.

circle inscribed in equilateral triangle

Strategy

The area of the triangle is comprised of the area of 6 triangles created by drawing the radii to the tangent points, and connecting the triangle's vertices to the circle's center. Let's draw two of those triangles:

The two radii, OD and OE, are equal to each other. From the Two Tangent theorem, we know that BD=BE, so ΔBOD is congruent to ΔBOE by the Side-Side-Side postulate. That means that ∠OBE≅∠OBD, and since m∠ABC=60° as an angle of an equilateral triangle, m∠OBD=30°.

The radii are perpendicular to the sides of the triangle (AB and BC) at the point where they meet the tangent, so ΔBOD is a 30-60-90 triangle. So BD=r·√3, as the long leg of a 30-60-90 right triangle where the short leg is r.

The area of each one of the small triangles is given by Asmall=base·height/2, or Asmall=r·r·√3/2=r2·√3/2. We have six of these triangles in the equilateral triangle, so its area is Aequilateral= 6·Asmall=6·r2√3/2=3r2√3.

Solution

(1) OE = OD = r //radii of a circle are all equal to each other
(2) BE=BD // Two Tangent theorem
(3) OB=OB //Common side, reflexive property of equality
(4) ΔBOD ≅ ΔBOE //Side-Side-Side postulate
(5) ∠OBE≅∠OBD //corresponding angles in congruent triangles
(6) m∠ABD = 60° //Given, ΔABC is equilateral
(7) m∠OBD = 30° // (5) , (6)
(8) m∠ODB=∠OEB=90° //radii are perpendicular to tangent line
(9) ΔBOD is a 30-60-90 triangle //(7), (8)
(10) BD= r·√3 //Properties of 30-60-90 triangle
(11) AΔBOD=base·height/2=r·r·√3/2=r2·√3/2
(12) AΔABC=6·AΔBOD=6·r2√3/2=3r2√3

« Another Converse Midsegment Theorem
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About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

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