How to Calculate the Area of an Isosceles Triangle

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An isosceles triangle has two equal sides. The area is the total space within the triangle. Whether you're trying to determine how much mulch to put in a triangular flower bed, how much paint you will need to cover the front of an A-line building, or simply drilling to hone your skills, plug what you do know into the triangle area formula.

The Formula

To find the area of an isosceles triangle, multiply the base, or width at the bottom of the triangle, and the height at tits tallest point, then divide the product in half. The base is the bottom side, or the side that is not equal to the other two. The height is the distance from the tallest peak of the triangle, the point where both even sides meet, to the base. The formula is A = ½ x b x h, where b is the base, and h is the height.

Plug It In

Plug your values into the formula to find the area. Multiply the base and height, then divide by 2. For example, if the base of the triangle is 8, and the height is 9, your formula will be Area = (½)(8)(9) = 36. If the base is 7 and the height is 3, the area is (½)(7)(3). Divide 21 by 2 for an area of 10.5.

Pythagorean Theorem

You may have to find the base or the height using the Pythagorean Theorem. The two halves of the isosceles triangle form two right triangles. The line that represents the height divides the isosceles triangle in half from bottom to tip and creates a right angle with the base. If you look at one of these right triangles, the height from the isosceles triangle will be one of the legs, half of the isosceles base will be the other leg, and the side of the isosceles triangle will be the hypotenuse. The Pythagorean Theorem formula is a2 + b2 = c2, where a and b are the legs of a right triangle, and c is the hypotenuse. You can use it to find height by solving for a or b. You can use it to find the base if you solve for a or b. Multiply the base solution by 2 to get the whole base measurement because the leg of the right triangle is only half of the base of the isosceles triangle.

Pythagorean Application

To find the base of an isosceles triangle with a side length of 5 and a height of 4, plug these into and solve: a2 + 42 = 52. Simplified, a2+16=25, and a2*=9*, so the answer is 3. This 3 is only half of the base, so the total base would be 6. To find the area of this triangle: A = (½)(4)(6), so the area would be 12.

Special Isosceles Triangle

A special isosceles triangle has inside angles of 45, 45 and 90 degrees and the sides are specific ratios toward one another. The formula to find the area of a 45-45-90 triangle is A = s2 ÷ 2, where s is length of a side. Square one of the side lengths, then divide the product in half. For example, to find the area of a triangle with sides 5, 5, and 7, your formula would be: A = 52 ÷ 2 or 25 ÷ 12.5. Therefore, the area of this 45-45-90 triangle is 12.5.

References

About the Author

Hannah Richardson has a Master's degree in Special Education from Vanderbilt University and a Bacheor of Arts in English. She has been a writer since 2004 and wrote regularly for the sports and features sections of "The Technician" newspaper, as well as "Coastwach" magazine. Richardson also served as the co-editor-in-chief of "Windhover," an award-winning literary and arts magazine. She is currently teaching at a middle school.

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