Types of triangles and types of angles: a parent's guide

You cut a square sandwich corner to corner, and your fourth grader asks what kind of triangles those two halves are. The homework wants two names for each triangle and a missing angle, and acute, obtuse and isosceles all sound alike after school. The types of triangles and the types of angles fit on one page once you know the two questions behind every name: how many sides are equal, and how big is the largest angle?

What are the types of angles?

The five types of angles are acute, right, obtuse, straight and reflex, and each one is defined by its size in degrees. Common Core standard 4.MD.C.5 defines the degree itself: "An angle that turns through 1/360 of a circle is called a 'one-degree angle,'" so a full turn is 360°.

• Acute angle: less than 90°. The hands of a clock at 1:00 or 2:00 make acute angles (30° and 60°).
• Right angle: exactly 90°, a square corner. The clock hands at 3:00 make one.
• Obtuse angle: more than 90° and less than 180°. The clock hands at 4:00 make 120°.
• Straight angle: exactly 180°, a straight line. The clock hands at 6:00 point in opposite directions.
• Reflex angle: more than 180° and less than 360°. The outside of the 4:00 angle, the long way round, is 360° − 120° = 240°.

A door shows the same ladder: opened a crack, it makes an acute angle with its frame; opened square, a right angle.

How to tell an angle's type without a protractor

The corner of a sheet of printer paper is a right angle, and it sorts any angle in a second. Slide the paper corner into the angle: if the angle hides inside the corner, it is acute; if the two match, it is right; if the angle opens wider than the paper, it is obtuse.

The national curriculum programmes of study ask Year 3 pupils to "identify whether angles are greater than or less than a right angle", and Year 5 pupils to "estimate and compare acute, obtuse and reflex angles".

How to classify a triangle, step by step

A triangle is classified by its sides and by its angles, so a full name has two words. Four steps give both.

1. Count the equal sides. Three equal sides make an equilateral triangle, two equal sides make an isosceles triangle, and no equal sides make a scalene triangle. On a drawing, equal sides carry the same number of small tick marks.
2. Find the largest angle. The largest angle sits opposite the longest side. Check it with the paper corner from the section above.
3. Name the angle type. If the largest angle is less than 90°, the triangle is acute; if it is exactly 90°, the triangle is right; if it is more than 90°, the triangle is obtuse.
4. Put the two names together. Your square sandwich, cut corner to corner, gives two right isosceles triangles: two equal sides, angles of 90°, 45° and 45°. A rectangular sandwich cut the same way gives two right scalene triangles.

The drawing shows the six names in order. The top row, left to right, is equilateral, isosceles and scalene; the bottom row is acute (largest angle 70°), right (90°) and obtuse (120°).

The part people get wrong

A triangle can have only one right angle or one obtuse angle, and children often draw a "triangle" with two. Two right angles already add up to 90° + 90° = 180°, the total for the whole triangle, so the third angle would be 0° and the sides would run parallel forever.

The same fact means every triangle has at least two acute angles, which is why the name depends on the largest angle only.

Types of triangles side by side

The six types of triangles split into two families of three, one by sides and one by angles. Every triangle belongs to one name from each family.

Some combinations never happen. An equilateral triangle is always acute, because its angles are all 60°, so a "right equilateral" or "obtuse equilateral" triangle cannot exist. Right isosceles (90°, 45°, 45°) and obtuse isosceles (120°, 30°, 30°) are both real.

Which names your child needs in which grade

Grade 4 is where the angle names and the first triangle category arrive in the US. Common Core 4.G.A.1 asks children to draw "angles (right, acute, obtuse)", and 4.G.A.2 asks them to "Recognize right triangles as a category, and identify right triangles."

Grade 5 adds the idea that categories nest. Standard 5.G.B.3 says that "attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category", and 5.G.B.4 asks children to "Classify two-dimensional figures in a hierarchy based on properties." Complementary and supplementary angles come in grade 7 (7.G.B.5), and the formal argument for the 180° angle sum waits for grade 8 (8.G.A.5).

In England, Year 4 pupils "compare and classify geometric shapes, including quadrilaterals and triangles, based on their properties and sizes", and Year 6 pupils "find unknown angles in any triangles, quadrilaterals, and regular polygons". See what math each grade should know for the rest of each year.

The 180° rule and missing angles

The three angles of every triangle add up to 180°. OpenStax Prealgebra states it as "For any ΔABC, the sum of the measures of the angles is 180°."

A missing angle is 180° minus the two angles you know. With angles of 50° and 60°, the third angle is 180° − 50° − 60° = 70°. Check it by adding: 50° + 60° + 70° = 180°.

A right triangle needs only one known angle. The right angle takes 90°, so a right triangle with a 35° angle has a third angle of 180° − 90° − 35° = 55°.

Two more angle facts use the same subtraction. Complementary angles add up to 90°, so the complement of 35° is 90° − 35° = 55°; supplementary angles add up to 180°, so the supplement of 50° is 180° − 50° = 130°. OpenStax defines them the same way: "If the sum of the measures of two angles is 180°, then the angles are supplementary."

Finding the third angle is two-digit subtraction, often with regrouping, as in 180 − 65 = 115. The guide to subtraction with regrouping helps when the zero in 180 causes trouble.

Play free in your browserSubtracting 2-Digit NumbersGrade 1–2

The tear-the-corners test proves the 180° rule at home. Draw any triangle, color its corners, cut it out, tear off the three corners and line their points up against a ruler: they always fill a straight line, which is 180°. Common Core 8.G.A.5 suggests the same idea: "arrange three copies of the same triangle so that the sum of the three angles appears to form a line".

How to find the area of a triangle

The area of a triangle is half of the base times the height: A = ½ × b × h, which is the same as (b × h) ÷ 2. The height is measured at a right angle from the base to the opposite corner, so it equals a side only in a right triangle.

The formula comes from a rectangle. Cut a rectangle 8 cm by 5 cm corner to corner like the sandwich, and each half is a triangle with a base of 8 cm and a height of 5 cm. The rectangle covers 8 × 5 = 40 cm², so each triangle covers 40 ÷ 2 = 20 cm².

OpenStax writes the formula as A = ½bh, "where b is the base and h is the height". Halving and multiplying by ½ are the same step (see fractions for parents). Common Core places triangle area in grade 6 (6.G.A.1), after the rectangle work in area and perimeter.

How to check progress with triangles and angles

Five quick questions show whether a child knows the names or is guessing.

• The clock question: What angle do the hands make at 3:00, 6:00 and 2:00? Right (90°), straight (180°) and acute (60°).
• The two-name question: What are the two names of a triangle with angles of 90°, 45° and 45°? Right isosceles.
• The missing angle: A triangle has angles of 40° and 75°. The third is 180° − 40° − 75° = 65°.
• The impossible triangle: Can a triangle have angles of 100° and 90°? No, because 100° + 90° = 190° is already more than 180°.
• The area question: A triangle has a base of 10 cm and a height of 6 cm. The area is 10 × 6 ÷ 2 = 30 cm².

A five-minute kitchen-table test

A five-minute kitchen-table test works best with real objects. Fold a napkin, cut a slice of toast, hold up a coat hanger, and ask for two names for each triangle, with the paper corner as the only tool.

If the names are solid but the missing angles stall, the gap is subtraction from 180. The math game for 4th graders drills that grade's number work, and the math game for parents and kids turns practice into a match you can play together.

Common mistakes and edge cases

Four mistakes cause most wrong answers on triangle and angle questions.

• Naming by looks alone: A triangle that looks right is called right without a check. Use the paper corner or the given angle, because 85° and 90° look alike.
• Reading "acute triangle" as one acute angle: Every triangle has at least two acute angles. An acute triangle needs all three under 90°.
• Subtracting from 360°: A child finds a missing triangle angle as 360° − 50° − 60° = 250°. Angles in a triangle add up to 180°; 360° is the total around a point.
• Using the slanted side as the height: In an acute or obtuse triangle, the height is a straight line at 90° to the base, which is shorter than the slanted side.

Is an equilateral triangle also isosceles?

An equilateral triangle is also isosceles under the inclusive definition, which says an isosceles triangle has at least two equal sides. Textbooks differ: some define isosceles as exactly two equal sides, and OpenStax says "An isosceles triangle has two sides the same length." Check which definition your child's textbook uses. The inclusive reading matches the grade 5 idea that a subcategory keeps every property of its category.

Your triangles and angles checklist and next steps

• Paper corner: keep a sheet of paper handy as a right-angle checker.
• Two questions: for every triangle, count the equal sides, then find the largest angle.
• Two names: say both, such as right scalene or obtuse isosceles.
• 180° check: add the three angles after every missing-angle answer.
• Area halving: multiply base by height, then halve it, and write the answer in square units.
• Word problems: when an angle question comes as a story, the steps in how to solve math word problems help pick the operation.

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and a short daily practice that keeps subtraction, multiplication and halving quick for missing-angle and area problems. Children can also challenge 1 to 4 friends or family members in real-time math battles, in any of 33 languages.