Tell the calculator which two things you know about a right-angled triangle: two sides, a side and an angle, or a side and the area. It works out everything else and draws the triangle to scale with each part labelled.
Choose which two values you know.
Type them in the boxes that appear.
Read the sides, angles, area and perimeter.
Check the drawing and follow the steps.
Sides: a² + b² = c²
Angles: sin α = a ÷ c, cos α = b ÷ c, tan α = a ÷ b, α + β = 90°
Area: ½ × a × b
Height on the hypotenuse: h = a × b ÷ c
A right triangle calculator finds every measurement of a right-angled triangle when you give it only two. A right-angled triangle is one in which two sides meet at 90°, like the corner of a book. The calculator returns the three sides, the two remaining angles, the area, the perimeter and the height drawn onto the longest side.
You are free to start from whichever pair of values you have. That may be the two shorter sides, one short side with the hypotenuse, one side with an angle, or one side with the area. The triangle is then drawn in true proportion with all parts named. This helps in trigonometry homework and also in practical jobs such as setting a ladder, cutting a rafter or planning a ramp.
1. Name the sides: a and b meet at the right angle, c is the longest side opposite it.
2. Note which two values you already have.
3. If you have two sides, use a² + b² = c² to get the third.
4. If you have a side and an angle, use sin, cos or tan to get the other sides.
5. Find one small angle with inverse tan of a ÷ b, and subtract it from 90° for the other.
6. Work out the area as half of a × b and the perimeter as a + b + c.
A triangle has three sides and three angles, six measurements in all. In a right triangle one angle is already fixed at 90°, and the three sides are tied together by the Pythagoras theorem. This leaves only two free choices. Once you fix two values, with at least one of them a length, the triangle is completely decided and no other shape can fit them.
For an acute angle in a right triangle, sine is the opposite side divided by the hypotenuse, cosine is the adjacent side divided by the hypotenuse, and tangent is the opposite side divided by the adjacent side. Many students remember this with a short phrase made from the first letters. These three ratios allow you to move from an angle to a side, or from two sides back to an angle.
Two right triangles appear again and again in exam questions. In the 45-45-90 triangle both short sides are equal and the hypotenuse is √2 times a short side. In the 30-60-90 triangle the hypotenuse is twice the shortest side, and the middle side is √3 times the shortest. Knowing these ratios lets you write the answer directly, with no need for trigonometric tables.
A painter in Nagpur leans a 5 metre ladder against a wall so that it makes an angle of 60° with the wall side opposite the ground reach, and wants to know how high it reaches. He enters the hypotenuse and the angle.
sin α = a ÷ c, so a = c × sin α: a = 5 × sin 60°
cos α = b ÷ c, so b = c × cos α: b = 5 × cos 60°
The three sides: a = 4.330127019, b = 2.5, c = 5
Angles: tan α = a ÷ b, and the two small angles add up to 90°: α = 60°, β = 90° − α = 30°
Area = ½ × a × b: ½ × 4.330127019 × 2.5 = 5.412658774
Height on the hypotenuse = a × b ÷ c: 4.330127019 × 2.5 ÷ 5 = 2.165063509
Answer: Sides a, b, c 4.3301, 2.5, 5; Angle α (opposite a) 60°; Angle β (opposite b) 30°
Taking a side other than the longest one as the hypotenuse.
Using sin where cos is needed, by mixing up the opposite and adjacent sides.
Keeping a hand calculator in radian mode while entering degrees.
Entering a hypotenuse shorter than the other side, which cannot form a triangle.
Forgetting that the two small angles must add up to exactly 90°.
Trigonometry problems on heights and distances in Class 10.
Deciding how long a ladder is needed to reach a given height.
Finding the slope length of a roof or a ramp.
Setting out a true right angle on a building site with a 3-4-5 triangle.
Checking the screen size of a TV from its width and height.
Can I solve a right triangle with only the two angles?
No. Angles fix the shape but not the size. You need at least one side.
What is a 30-60-90 triangle?
A right triangle with angles 30°, 60° and 90°. Its sides are always in the ratio 1 : √3 : 2, so the hypotenuse is double the shortest side.
What is a 45-45-90 triangle?
A right triangle with two equal sides. Its sides are in the ratio 1 : 1 : √2. It is half of a square cut along the diagonal.
How do I find an angle from two sides?
Divide the side opposite the angle by the side next to it and take the inverse tan. For sides 3 and 4, the angle is tan⁻¹(3 ÷ 4), about 36.87°.
What is the height on the hypotenuse?
It is the perpendicular drawn from the right-angle corner to the hypotenuse. Its length is a × b ÷ c.