Area of isosceles triangle formula with sides without height

First draw the height - due to the symmetry of equilateral triangles, it will start at the midpoint of whichever side you choose to start from, and end at the opposite vertex (point/corner). In effect you will split the equilateral triangle into two congruent right triangles. Now consider one of these two right triangles by itself. If the equilateral triangle has sides of length x, then the hypotenuse of our right triangle will also be x. We also know that the side opposite the 30 degree angle has length x/2, since we split that side of the triangle in half to construct this right triangle. Since you know two of the sides of a right triangle, you can use the Pythagorean theorem to find the length of the 3rd. (x/2)^2 + m^2 = x^2 x^2/4 + m^2 = x^2 m^2 = (3*x^2)/4 m = (x*sqrt(3))/2 Where m is the height of the right triangle, which is equal to the height of the equilateral triangle. Incidentally, this derivation also proves the shortcut for the ratio of the sides in a 30-60-90 triangle, since the effect of cutting an equilateral triangle in half is to create 2 30-60-90 triangles. We don't know what the height is without working it out. Pythagoras' theorem says that in a right triangle, the area of a square drawn on the longest side (the hypotenuse) is equal in area to the areas of the two squares drawn on the other two sides (the legs), added together. The square on the side with length s/2 is 1/4 of the area of the square on the side with length s, so a square on the height ...

An isosceles triangle is a and the remaining side has length . This property is equivalent to two angles of the triangle being equal. An isosceles triangle therefore has both two equal sides and two equal angles. The name derives from the Greek iso (same) and skelos ( A triangle with all sides equal is called an but all three sides and angles equal. Another special case of an isosceles triangle is the The height of the isosceles triangle illustrated above can be found from the More things to try: • • • References Gearhart, W.B. and Schulz, H.S. "The Function ." College Math. J. 21, 90-99, 1990. Cite this as: MathWorld--A Wolfram Web Resource. Subject classifications • • • • • • • • • • • • • • • • • • • • Created, developed and nurtured by Eric Weisstein at Wolfram Research

It won't be easy but if you look carefully at the isosceles triangle it's a 45, 45, 90 triangle when split in half And to find the hypotenuse you have to multiply by the square root of 2 but we are not trying to find the hypotenuse we are trying to find the height So we have to do the opposite instead of multiplying by the square root of 2 you have to divide by the square root of 2 So we already know the hypotenuse which is 13 so it would be (13/√2) usually we can leave it like this but we can also rationalize it by multiplying (13/√2) with (√2/√2) which is approximately 9.19 Hopefully you found that helpful :) don't forget to vote! (And in case you are wondering why the height is not the same is because the drawing in the video is not up to scale if the hypotenuse is 13 then really if you want to be exact then 9.19 is probably your best bet but now you should just roll with it) - [Instructor] We're asked to find the value of x in the isosceles triangle shown below. So that is the base of this triangle. So pause this video and see if you can figure that out. Well the key realization to solve this is to realize that this altitude that they dropped, this is going to form a right angle here and a right angle here and notice, both of these triangles, because this whole thing is an isosceles triangle, we're going to have two angles that are the same. This angle, is the same as that angle. Because it's an isosceles triangle, this 90 degrees is the same as that 90 degrees. And so...

Isosceles Triangle Shape A = angle A a = side a B = angle B b = side b C = angle C c = side c A = C a = c ha = hc K = area P = perimeter See Diagram Below: ha = altitude of a hb = altitude of b hc = altitude of c *Length units are for your reference only since the value of the resulting lengths will always be the same no matter what the units are. Calculator Use An isosceles triangle is a special case of a In our calculations for a right triangle we only consider 2 known sides to calculate the other 7 unknowns. For example, if we know a and b we know c since c = a. Once we know sides a, b, and c we can calculate the perimeter = P, the semiperimeter = s, the area = K, and the altitudes: ha, hb, and hc. Formulas and Calculations for an isosceles triangle: • Sides of Isosceles Triangle: a = c • Angles of Isosceles Triangle: A = C • Altitudes of Isosceles Triangle: ha = hc • Perimeter of Isosceles Triangle: P = a + b + c = 2a + b • Semiperimeter of Isosceles Triangle: s = (a + b + c) / 2 = a + (b/2) • Area of Isosceles Triangle: K = (b/4) * √(4a 2 - b 2) • Altitude a of Isosceles Triangle: ha = (b/2a) * √(4a 2 - b 2) • Altitude b of Isosceles Triangle: hb = (1/2) * √(4a 2 - b 2) • Altitude c of Isosceles Triangle: hc = (b/2a) * √(4a 2 - b 2) Calculation: Given sides a and b find side c and the perimeter, semiperimeter, area and altitudes • a and b are known; find c, P, s, K, ha, hb, and hc • c = a • P = 2a + b • s = a + (b/2) • K = (b/4) * √(4a 2 - b 2) • ha = (b/2a) * √(4a 2 ...

Area = 14.8 to one decimal place How to Remember Just think "abc": Area = ½ a b sin C It is also good to remember that the angle is always between the two known sides, called the "included angle". How Does it Work? We start with this formula: Area = ½ × base × height We know the base is c, and can work out the height: the height is b × sin A So we get: Area = ½ × (c) × (b × sin A) Which can be simplified to: Area = 1 2bc sin A By changing the labels on the triangle we can also get: • Area = ½ ab sin C • Area = ½ ca sin B One more example: Example: Find How Much Land Farmer Rigby owns a triangular piece of land. The length of the fence AB is 150 m. The length of the fence BC is 231 m. The angle between fence AB and fence BC is 123º. How much land does Farmer Rigby own? First of all we must decide which lengths and angles we know: • AB = c = 150 m, • BC = a = 231 m, • and angle B = 123º So we use: Area = 1 2ca sin B

First draw the height - due to the symmetry of equilateral triangles, it will start at the midpoint of whichever side you choose to start from, and end at the opposite vertex (point/corner). In effect you will split the equilateral triangle into two congruent right triangles. Now consider one of these two right triangles by itself. If the equilateral triangle has sides of length x, then the hypotenuse of our right triangle will also be x. We also know that the side opposite the 30 degree angle has length x/2, since we split that side of the triangle in half to construct this right triangle. Since you know two of the sides of a right triangle, you can use the Pythagorean theorem to find the length of the 3rd. (x/2)^2 + m^2 = x^2 x^2/4 + m^2 = x^2 m^2 = (3*x^2)/4 m = (x*sqrt(3))/2 Where m is the height of the right triangle, which is equal to the height of the equilateral triangle. Incidentally, this derivation also proves the shortcut for the ratio of the sides in a 30-60-90 triangle, since the effect of cutting an equilateral triangle in half is to create 2 30-60-90 triangles. We don't know what the height is without working it out. Pythagoras' theorem says that in a right triangle, the area of a square drawn on the longest side (the hypotenuse) is equal in area to the areas of the two squares drawn on the other two sides (the legs), added together. The square on the side with length s/2 is 1/4 of the area of the square on the side with length s, so a square on the height ...

An isosceles triangle is a and the remaining side has length . This property is equivalent to two angles of the triangle being equal. An isosceles triangle therefore has both two equal sides and two equal angles. The name derives from the Greek iso (same) and skelos ( A triangle with all sides equal is called an but all three sides and angles equal. Another special case of an isosceles triangle is the The height of the isosceles triangle illustrated above can be found from the More things to try: • • • References Gearhart, W.B. and Schulz, H.S. "The Function ." College Math. J. 21, 90-99, 1990. Cite this as: MathWorld--A Wolfram Web Resource. Subject classifications • • • • • • • • • • • • • • • • • • • • Created, developed and nurtured by Eric Weisstein at Wolfram Research

Isosceles Triangle Shape A = angle A a = side a B = angle B b = side b C = angle C c = side c A = C a = c ha = hc K = area P = perimeter See Diagram Below: ha = altitude of a hb = altitude of b hc = altitude of c *Length units are for your reference only since the value of the resulting lengths will always be the same no matter what the units are. Calculator Use An isosceles triangle is a special case of a In our calculations for a right triangle we only consider 2 known sides to calculate the other 7 unknowns. For example, if we know a and b we know c since c = a. Once we know sides a, b, and c we can calculate the perimeter = P, the semiperimeter = s, the area = K, and the altitudes: ha, hb, and hc. Formulas and Calculations for an isosceles triangle: • Sides of Isosceles Triangle: a = c • Angles of Isosceles Triangle: A = C • Altitudes of Isosceles Triangle: ha = hc • Perimeter of Isosceles Triangle: P = a + b + c = 2a + b • Semiperimeter of Isosceles Triangle: s = (a + b + c) / 2 = a + (b/2) • Area of Isosceles Triangle: K = (b/4) * √(4a 2 - b 2) • Altitude a of Isosceles Triangle: ha = (b/2a) * √(4a 2 - b 2) • Altitude b of Isosceles Triangle: hb = (1/2) * √(4a 2 - b 2) • Altitude c of Isosceles Triangle: hc = (b/2a) * √(4a 2 - b 2) Calculation: Given sides a and b find side c and the perimeter, semiperimeter, area and altitudes • a and b are known; find c, P, s, K, ha, hb, and hc • c = a • P = 2a + b • s = a + (b/2) • K = (b/4) * √(4a 2 - b 2) • ha = (b/2a) * √(4a 2 ...

It won't be easy but if you look carefully at the isosceles triangle it's a 45, 45, 90 triangle when split in half And to find the hypotenuse you have to multiply by the square root of 2 but we are not trying to find the hypotenuse we are trying to find the height So we have to do the opposite instead of multiplying by the square root of 2 you have to divide by the square root of 2 So we already know the hypotenuse which is 13 so it would be (13/√2) usually we can leave it like this but we can also rationalize it by multiplying (13/√2) with (√2/√2) which is approximately 9.19 Hopefully you found that helpful :) don't forget to vote! (And in case you are wondering why the height is not the same is because the drawing in the video is not up to scale if the hypotenuse is 13 then really if you want to be exact then 9.19 is probably your best bet but now you should just roll with it) - [Instructor] We're asked to find the value of x in the isosceles triangle shown below. So that is the base of this triangle. So pause this video and see if you can figure that out. Well the key realization to solve this is to realize that this altitude that they dropped, this is going to form a right angle here and a right angle here and notice, both of these triangles, because this whole thing is an isosceles triangle, we're going to have two angles that are the same. This angle, is the same as that angle. Because it's an isosceles triangle, this 90 degrees is the same as that 90 degrees. And so...

Area = 14.8 to one decimal place How to Remember Just think "abc": Area = ½ a b sin C It is also good to remember that the angle is always between the two known sides, called the "included angle". How Does it Work? We start with this formula: Area = ½ × base × height We know the base is c, and can work out the height: the height is b × sin A So we get: Area = ½ × (c) × (b × sin A) Which can be simplified to: Area = 1 2bc sin A By changing the labels on the triangle we can also get: • Area = ½ ab sin C • Area = ½ ca sin B One more example: Example: Find How Much Land Farmer Rigby owns a triangular piece of land. The length of the fence AB is 150 m. The length of the fence BC is 231 m. The angle between fence AB and fence BC is 123º. How much land does Farmer Rigby own? First of all we must decide which lengths and angles we know: • AB = c = 150 m, • BC = a = 231 m, • and angle B = 123º So we use: Area = 1 2ca sin B