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Isosceles Triangle Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Result

12.0000 cm²

Area

Perimeter
16.0000 cm
Height
4.0000 cm
Apex angle
73.74 °
Base angle
53.13 °

An isosceles triangle calculator takes a base and a leg and returns five measurements: the area, the perimeter, the height, the apex angle and the base angle. It sits between the other two triangle pages and the reason is arithmetic. A general triangle has three degrees of freedom — three sides, or two sides and the angle between them — and needs that many numbers before the shape is fixed. An equilateral triangle has none, because the shape is settled in advance and only the size is left. An isosceles triangle has exactly two: the base and the leg, and once both are given there is nothing further to choose. Two legs of a stated length can only meet at one point above a base of a stated length, so the height, both angles and the area all follow. The shape is the one you get by folding a triangle in half down the middle, and that fold is also the way to work it out: cutting it along its axis of symmetry gives two identical right-angled triangles, each with a hypotenuse equal to the leg, a base of half the base, and a height given by Pythagoras. Every output on the page comes from that one right-angled triangle. The height is the third side of it, the area is the base times that height halved, and the two angles come from the same triangle as well — half the apex angle is the angle whose sine is half the base over the leg, and the base angle is what is left of 180 degrees after the apex is taken out, split evenly between the two of them. That the two base angles are equal is the definition of the shape rather than a result, and it is the thing a drawing usually gives you: mark one base angle and the other is known. The page prints both angles because they answer different questions — the apex angle is what tells you how pointed the shape is, and the base angle is what a mitre or a roof pitch is actually quoted in.

Isosceles triangles from the default shape to the flat boundary case

Base (cm)Leg (cm)Area (cm²)Height (cm)Apex angle (°)Base angle (°)
6512473.7453.13
85123106.2636.87
1013601245.2467.38
6615.58855.19626060
434.47212.236183.6248.19
0505090
63001800

Seven triangles, and the first row is the one the page loads with. The perimeter is deliberately not a column here: it is the base plus twice the leg, which the first two columns already give, so a column for it would be a third number the reader can add up in their head. Two rows are cross-checks against the pages beside this one, and both of them land on the same figures from a different set of inputs: the first row, a base of 6 with legs of 5, gives an area of 12 and a perimeter of 16, which is what the Heron page returns for sides of 6, 5 and 5; and the fourth row, a base and legs of 6, gives 15.5885, which is what the equilateral page returns for a side of 6 and the Heron page for 6, 6, 6. The fourth row is also the only one on the table where the two angle columns are equal, both 60, which is the signature of an equilateral triangle and the reason the page prints the two angles separately everywhere else. Read the apex and base angle columns together as the base grows: at a base of 4 with legs of 3 the apex is 83.62, at the default 6 with legs of 5 it is 73.74, at 8 with legs of 5 it is 106.26 and at 10 with legs of 13 it is 45.24 — the two columns always add up with the apex to 180 when the apex is counted once and the base angle twice, which is the check to run if a row looks wrong. The last two rows are the boundary cases and both are real answers rather than failures: a base of 0 with legs of 5 is a doubled line of length 5 with an apex angle of 0, and a base of 6 with legs of 3 has flattened completely, giving an apex angle of 180 and an area of 0. The second of those is the exact edge — one more tenth on the base and the page refuses. Every number here is recomputed from its base and leg when the page is built, in centimetres and degrees, and the decimal widths are the same ones the results panel uses, four for the lengths and two for the angles.

Formula

h = √(l² − (b ÷ 2)²) A = b·h ÷ 2 P = b + 2l apex = 2·asin((b ÷ 2) ÷ l) base = (180° − apex) ÷ 2

Base
The length of the unequal side, the one the two legs stand on, in centimetres. Laying it flat is what makes the triangle symmetrical left to right, and it is the side the height is measured from
Leg
The length of each of the two equal sides, in centimetres. Both are the same by definition, so there is only one box for them, and each leg is the hypotenuse of the right-angled triangle the fold produces
Area
The space the triangle covers, in square centimetres: half the base times the height. The area is worked out from the base and the leg directly rather than from the printed height, so the panel's own figures agree with each other to the last digit
Perimeter
The distance all the way around, in centimetres: the base plus twice the leg. It is the one output that needs no square root and no trigonometry, so it is exact whenever the two inputs are
Height
The distance from the apex straight down to the middle of the base, in centimetres, from Pythagoras on the folded half: the leg squared minus half the base squared, square-rooted. It is also the line the triangle is symmetric about, which is why it lands exactly at the middle of the base and not off to one side
Apex angle
The angle between the two legs, at the top, in degrees, which is the one angle of an isosceles triangle that is not shared with another. It is twice the angle whose sine is half the base over the leg, and it approaches 180 degrees as the triangle flattens
Base angle
The angle between the base and a leg, in degrees. The two base angles are equal by definition, so the page prints one figure for both, worked out as what remains of 180 degrees once the apex angle is taken away, halved
Two decimal places for the angles
The angles print two decimals where the lengths print four, because a protractor reads to about a hundredth of a degree and the trailing digits of an arcsine are floating-point noise rather than anything anyone measured

The page is for anything built on an isosceles triangle, and that shape turns up wherever a symmetrical point sits on a flat base. A roof is the clearest case: the two slopes are legs, the span is the base, and what a carpenter wants is the pitch — the base angle — and the height of the ridge above the wall plate, which is the height on this page. Cut that triangle down the middle and you have the rafter length as the hypotenuse, which is why the fold is the working method rather than a trick of the diagram. Gable ends, pediments, arrowheads, funnel cones and the front of a tent are the same shape with different names. In metalwork and joinery the base angle is what a mitre saw is set to, and the apex angle is what you check the finished piece against. The page is also the right one when you have measured a base and one sloping side rather than a height: on a roof, a ramp or a triangular brace, the sloped edge is what you can reach and the perpendicular is not. And it is the page to reach for when a drawing dimensions an isosceles triangle on its symmetry — a base and an overall height, say — since the height here is an output rather than an input, and the two are not interchangeable: base and leg is one pair of inputs, base and height is another, and only one of the two is on this page. If your triangle is not isosceles, the triangle area page takes a base and a height and the Heron page takes three sides.

Worked examples

  1. A base of 6 with legs of 5

    1. Half the base: 6 ÷ 2 = 3
    2. Height: √(25 − 9) = √16 = 4
    3. Area: 6 × 4 ÷ 2 = 12
    4. Perimeter: 6 + 2 × 5 = 16
    5. Half the apex angle: asin(3 ÷ 5) = 36.8699…, so the apex angle is 73.7398…, which rounds to 73.74
    6. Base angle: (180 − 73.7398) ÷ 2 = 53.1301…, which rounds to 53.13

    The input the page loads with, and the cleanest one on it: the height comes out at exactly 4, the area at exactly 12 and the perimeter at exactly 16, with only the two angles leaving a decimal behind. These are the sides 6, 5 and 5, so this is the second of the cross-checks in this batch — the Heron page takes those three sides and returns the same area of 12 and the same perimeter of 16. Two routes to one triangle, one answer. The base angle of 53.13 degrees is not a coincidence either: folding this triangle leaves a 3-4-5 right-angled triangle, and 53.13 is the angle a rise of 4 over a run of 3 makes with the horizontal — which is the rule the page's base angle column follows for every row.

  2. A base of 6 with legs of 6

    1. Half the base: 6 ÷ 2 = 3
    2. Height: √(36 − 9) = √27 = 5.196152…, which rounds to 5.1962
    3. Area: 6 × 5.196152… ÷ 2 = 15.588457…, which rounds to 15.5885
    4. Perimeter: 6 + 2 × 6 = 18
    5. Half the apex angle: asin(3 ÷ 6) = 30, so the apex angle is 60 and the base angle is (180 − 60) ÷ 2 = 60

    An isosceles triangle whose base and legs all happen to be 6, which makes it equilateral — and the page shows that rather than hiding it: both angles come out at 60 degrees, which is the signature of the shape. The area of 15.5885 is the third of the cross-checks in this batch, matching the equilateral page at a side of 6 and the Heron page at 6, 6, 6. This is also the only row on the table where the two angles are equal, since the apex angle only equals the base angle when all three are 60.

  3. A base of 8 with legs of 5

    1. Half the base: 8 ÷ 2 = 4
    2. Height: √(25 − 16) = √9 = 3
    3. Area: 8 × 3 ÷ 2 = 12
    4. Perimeter: 8 + 2 × 5 = 18
    5. Half the apex angle: asin(4 ÷ 5) = 53.1301…, so the apex angle is 106.2602…, which rounds to 106.26
    6. Base angle: (180 − 106.2602) ÷ 2 = 36.8699…, which rounds to 36.87

    A flatter triangle than the default — a longer base with the same legs — and the two angles move in opposite directions, which is the thing to watch. The apex angle has gone from 73.74 to 106.26 and each base angle from 53.13 to 36.87, and the two figures are still tied together by the rule that all three angles add to 180. The area has stayed at 12 while the perimeter has gone up to 18, which is a reminder that stretching a base flattens a triangle without making it cover more ground. The 3-4-5 right-angled triangle is hiding inside this one: half the base is 4, the height is 3 and the leg is 5.

  4. A base of 10 with legs of 13

    1. Half the base: 10 ÷ 2 = 5
    2. Height: √(169 − 25) = √144 = 12
    3. Area: 10 × 12 ÷ 2 = 60
    4. Perimeter: 10 + 2 × 13 = 36
    5. Half the apex angle: asin(5 ÷ 13) = 22.6198…, so the apex angle is 45.2397…, which rounds to 45.24
    6. Base angle: (180 − 45.2397) ÷ 2 = 67.3801…, which rounds to 67.38

    The second 5-12-13 half on this batch: half the base is 5, the height is 12 and the leg is 13, so the right-angled triangle inside is the same one the Heron page meets from the other direction. Everything except the two angles comes out whole. The apex angle of 45.24 degrees is a little over half a right angle, which is a useful sanity check on the shape: a triangle with a base of 10 and legs of 13 is close to but not quite an isosceles right triangle, and going to a base of 13 times the square root of two would make the apex exactly 90.

  5. A base of 0 with legs of 5

    1. Half the base: 0 ÷ 2 = 0
    2. Height: √(25 − 0) = 5
    3. Area: 0 × 5 ÷ 2 = 0
    4. Perimeter: 0 + 2 × 5 = 10
    5. Half the apex angle: asin(0 ÷ 5) = 0, so the apex angle is 0 and the base angle is (180 − 0) ÷ 2 = 90

    A base of zero, which is legal and is not the same as leaving the box empty. The two legs fall on top of each other, so the figure is a doubled line of length 5: the height is 5, the perimeter is 10, the area is zero and the apex angle has closed to nothing. The base angle of 90 degrees is the honest reading of that — the base has no length, so the angle between it and a leg is a right angle by convention. This is also why the page treats a zero base as a real input rather than refusing it: the arithmetic is all defined and the answers describe the collapsed shape correctly.

  6. A base of 6 with legs of 3

    1. Half the base: 6 ÷ 2 = 3
    2. Height: √(9 − 9) = 0
    3. Area: 6 × 0 ÷ 2 = 0
    4. Perimeter: 6 + 2 × 3 = 12
    5. Half the apex angle: asin(3 ÷ 3) = 90, so the apex angle is 180 and the base angle is (180 − 180) ÷ 2 = 0

    The exact boundary, and it is allowed through: the two legs laid end to end are exactly as long as the base, so the triangle has flattened into a straight line. The apex angle is 180 degrees, which is what a completely flat point measures, and the two base angles have closed to zero. Nothing here is a NaN — every term in the arithmetic is defined, including the arcsine of exactly 1. Push the base to 6.2 with the same legs and the page refuses, because then the two legs cannot reach each other at all and no triangle exists.

Limitations

The page is for isosceles triangles only: exactly two sides the same length, which are the legs, standing on the unequal base. A triangle that is scalene needs all three sides and belongs on the Heron page; one that is right-angled needs its own page, and an equilateral triangle is only a special case of this one when its base and its legs happen to be given as equal. The two legs must reach each other: if twice the leg is less than the base, no triangle exists and the page says so instead of returning a number. The exact case where twice the leg equals the base is allowed through and describes a triangle flattened into a line, with a height and an area of zero and an apex angle of 180 degrees. The page takes a base and a leg rather than a base and a height, so a drawing dimensioned the other way needs the height converted into a leg first, by Pythagoras on half the base. The outputs are in centimetres and square centimetres whatever the unit dropdown is set to, and the angles are in degrees with no radians option. Four decimals for the lengths and two for the angles is a display width rather than a claim about precision; on a triangle measured with a tape, the height and the area are the figures whose last digits have any chance of being meaningful, and even those depend on how well the base and the leg were measured. The page is two-dimensional and has no field for thickness, so it cannot give the volume of a triangular prism or the weight of a gusset.

Frequently asked questions

Why does the page ask for a leg instead of the height?
Because the base and the leg are the pair of numbers that fixes the shape, and the height follows from them. Two legs of a given length can only meet at one point above a base of a given length, so nothing is left to choose. A base and a height is a different pair of two numbers for the same triangle, and it would over-determine the shape if the page took both — a base of 6 with a height of 5 would force the legs to be the square root of 34, and the page would have to decide which of the three numbers you typed was wrong.
Why are there two angles, and are they ever the same?
The apex angle is between the two legs, at the top, and the base angle is between the base and a leg. They are separate quantities and they answer different questions: the apex angle says how pointed the shape is, the base angle is what a roof pitch or a mitre setting is quoted in. They are equal only in the equilateral case, where all three angles are 60 degrees — which is why the page prints both rather than one figure labelled angle.
How do I get the angles from the two sides?
Cut the triangle in half down its axis of symmetry. Each half is a right-angled triangle with a hypotenuse of the leg and a base of half the base, so the sine of half the apex angle is half the base divided by the leg. Double that and you have the apex angle; the base angle is whatever is left of 180 degrees after the apex is taken out, split in two. The page does exactly this, which is why half a base of 3 with a leg of 5 gives an apex angle of 73.74 rather than something you would guess.
Why do the angles show two decimals and the lengths four?
Because the two were measured to different standards. A length can be exact — a base of 6 is 6 — and four decimals is the width this site prints lengths at. An angle here comes out of an arcsine, and a protractor or a mitre saw reads to about a hundredth of a degree, so the third and fourth decimals of an angle would be floating-point noise dressed up as precision. Two is as far as the number means anything.
What happens if the legs are too short to reach across the base?
The page stops and tells you rather than returning a number. The two legs have to be able to meet above the base, which means their lengths added together must be greater than the base; if they are not, no triangle exists. The exact case where their sum equals the base is allowed through, and it describes a triangle flattened into a straight line with an apex angle of 180 degrees and an area of zero, which is a real answer rather than a failure.
Can the base or the leg be zero?
Both are accepted and both give meaningful answers. A leg of zero puts both legs at a point with the base stretched out flat, and a base of zero puts the two legs on top of each other, giving a height equal to the leg, a perimeter of twice the leg, and a base angle of 90 degrees. Neither is the same as leaving the box empty: a blank box shows no result at all, because there is nothing to work from, while a zero is a shape you can describe.

References

Related calculators

Triangle Area CalculatorHalf a base times the perpendicular height above it.Heron's Formula CalculatorTriangle area from three side lengths, by Heron's formula.Equilateral Triangle CalculatorArea, height and perimeter of an equilateral triangle.Right Triangle CalculatorSolve a right triangle from two sides or a side and angle.