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How To Calculate the Area of a Triangle
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Jun 28, 2025
In this video, I’ll guide you through the process of calculating the area of a triangle using two different methods. We’ll start with Heron’s formula for those familiar with it, and then we’ll explore an alternative method that doesn’t require it. Whether you’re brushing up on your geometry skills or tackling this problem for the first time, this step-by-step explanation will help you understand how to find the area of a triangle. Watch the video to see how both methods lead to the same result, and don’t forget to share your thoughts in the comments!
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0:00
hello welcome to we learn
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daily in this video we will find the
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area of a triangle with sides measuring
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13 14 and 15
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units as usual before watching try to
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solve this problem on your own let's
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start we will find the area of the
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triangle using two methods the first
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method is for those who know Heron's
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formula and the second method is for
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those who don't first meth method using
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Heron's formula the area of a triangle
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according to Heron's formula is equal to
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the square root of the product of the
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semi perimeter multiplied by the semi
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perimeter minus one of the sides
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multiplied by the semi perimeter minus
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the second side and multiplied by the
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semi perimeter minus the third side
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let's first find the semi perimeter of
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the triangle which is equal to half the
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sum of all three sides we have sides
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measuring 13 14 and 15
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so we add these together and divide by
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two the sum of the sides is 42 and
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dividing 42 by 2 gives us a semi
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perimeter of 21 now let's substitute the
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semi perimeter into the formula for the
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area of the triangle under the square
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root we have the product of 21
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multiplied 21 minus one of the sides for
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example 13 which equal 8 multiplied by
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21 minus the second side for example 14
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which equals 7 and multiplied by 21 - 15
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which equals 6 next we
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simplify 21 can be broken down into 7 *
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3 and 6 can be broken down into 3 * 2 so
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we have the product of 27s and the
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square < TK of 49 is 7 then we multiply
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this by the square root of the product
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of 2 3es which equals 3 and multiply by
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the square < TK of 16 which equals 4
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finally we multiply 7 by 3 which gives
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us 21 and multiply that by four to get
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84 square units second method using the
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height of the triangle let's label the
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vertices of the triangle with letters
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such as a b and c from vertex B we drop
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a height perpendicular to side AC and
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label this height
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BD now let's try to find this height
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let's denote segment a by the variable X
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the length of segment DC can be found as
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the difference between segments a c and
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a d so we get 15 - x the height BD is
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the leg of a right triangle ABD so
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according to the Pythagorean theorem the
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square of the height BD is equal to the
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square of the hypotenuse AB minus the
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square of the other leg a d on the other
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hand the square of the height BD can
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also be expressed using the Pythagorean
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theorem from right triangle
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BDC therefore bd^ 2 is equal to the
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square of the hypotenuse BC minus the
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square of the leg DC substituting X for
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a d and 15 - x for DC we get the
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following equation the square of side AB
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which is 132 - x^2 = the square of side
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BC which is 14^ 2us the square of 15 - x
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let's move everything containing the
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variable X to one side and the constants
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to the other
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side on the left side we have the
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difference of squares the square of 15 -
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x -
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x^2 on the right side we have the
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difference of squares the square of 14
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minus the square of 13 it's convenient
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to expand both the left and right sides
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using the difference of squares formula
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on the left side we get 15 - x - x * 15
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- x + x on the right side we get 14 -3 *
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14 + 13 simplifying on the left side we
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get 15 - 2 * X multiplied 15 and this is
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equal to
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27 from this equation we find that X =
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6.6 returning to the height BD we find
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it as the leg of the right triangle
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ABD the height BD is equal to the square
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root of the difference of the squares of
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side AB which is 13 squar and a d which
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is 6.6 squar the difference under the
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square root can also be calculated using
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the difference of squares formula so we
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get 13 - 6.6 * 13 + 6.6 under the square
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root we have 6.4 multiplied by
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19.6 to find the square root of this
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product it's convenient to express 6.4
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as 6410 and 19.6 as
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19610 taking the square root of the
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numerator we get the product of the
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square roots of 64 and
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196 which equals 8 *
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14 the square otk of 10^ s is 10 so we
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divide the result by 10 to get
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11210 which equals
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11.2 now we can find the area of the
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triangle which is equal to half the
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product of the the height BD and the
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base
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AC we multiply 1/2 by 11.2 and by 15 to
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get 168 ID 2 which equals 84 write the
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answer the area of the triangle is 84
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square units the problem is solved if
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the solution is clear please like the
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video leave a comment and don't forget
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to subscribe to the
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channel thank you for watching and let's
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see you in the next one goodbye
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