Question Video: Using Heron’s Formula to Find the Area of a Triangle Formed by Joining the Centers of Three Tangential Circles | Nagwa Question Video: Using Heron’s Formula to Find the Area of a Triangle Formed by Joining the Centers of Three Tangential Circles | Nagwa

# Question Video: Using Heron’s Formula to Find the Area of a Triangle Formed by Joining the Centers of Three Tangential Circles Mathematics • Second Year of Secondary School

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Three tangential circles have radii 200 cm, 120 cm, and 110 cm. Find the area of the triangle formed by connecting the centers of the circles. Give your answer to two decimal places.

02:45

### Video Transcript

Three tangential circles have radii 200 centimeters, 120 centimeters, and 110 centimeters. Find the area of the triangle formed by connecting the centers of the circles. Give your answer to two decimal places.

We’re given the radii of three tangential circles. So let’s begin by sketching the circles. We want to find the area of the triangle formed by connecting the centers of the circles. And to find the area of this triangle, we can use Heron’s formula. This says that for a triangle with side lengths 𝑎, 𝑏, and 𝑐, the area is equal to the square root of 𝑠 times 𝑠 minus 𝑎 times 𝑠 minus 𝑏 times 𝑠 minus 𝑐, where 𝑠 is the semiperimeter. That’s half the sum of the side lengths.

Our first step now is to work out the side lengths of our triangle. And these are each the sum of the radii of two circles. So, for example, calling our first side length 𝑎, this is the sum of the radius of the largest circle, 200 centimeters, and that of the second largest circle, 120 centimeters, which is 320 centimeters. Similarly, our next side 𝑏 is the sum of the largest radius, 200 centimeters, with the smallest radius, 110 centimeters. And that’s 310 centimeters. Our third side length 𝑐 of the triangle is the sum of the two smaller radii. That’s 120 plus 110 centimeters, which is 230 centimeters.

So, with our side lengths 320, 310, and 230 centimeters, we can now work out the semiperimeter 𝑠 to use in Heron’s formula. That’s 320, which is 𝑎, plus 310, which is 𝑏, plus 230, which is 𝑐, all over two. And that’s 860 over two, which is 430. So 𝑠 is equal to 430.

So now making some space, we can use the values we’ve calculated in Heron’s formula to find the area of the triangle. So we have the area 𝑎 is equal to the square root of 430 times 110 times 120 times 200. That’s the positive square root, since area is always positive, of 1,135,200,000, which is approximately equal to 33,692.72919 and which is 33,692.73 to two decimal places.

Hence, the area of the triangle formed by connecting the centers of the three tangential circles, to two decimal places, is 33,692.73 centimeters squared.

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