# Triangles: Definition, Types, Properties, and Examples **TL;DR** A triangle is a closed two-dimensional shape with three sides, three angles, and three vertices, and its interior angles always add to 180°. This guide covers a triangle's parts, the six types (by side and by angle), the core properties, area and perimeter formulas, six worked examples, and common mistakes. A **triangle** is a closed, flat (two-dimensional) figure made of **three straight sides** that meet at **three corners**. Those corners are the **vertices** (singular: vertex), the straight edges are the **sides**, and the space opened up at each vertex is an **interior angle**. Because the sides close the figure completely, a triangle always encloses a region — the region whose size we call its area. Every triangle, no matter how stretched or skewed, obeys the same rules. By the end of this guide you will be able to name a triangle's parts, sort any triangle into its type by side and by angle, apply the angle-sum and triangle-inequality properties, and compute area and perimeter. The deeper toolkit — the full set of [properties of a triangle](/content/math/geometry/properties-of-a-triangle/index.html) and the formal [triangle sum theorem](/content/math/geometry/triangle-sum-theorem/index.html) — builds on what follows. ## Parts of a Triangle Geometry uses a tidy labeling convention so a diagram reads the same to everyone: - **Vertices** — the three corner points, labelled with capital letters: A, B, C. - **Sides** — the three edges, labelled with lowercase letters: side a, b, c, where each side is named for the vertex _opposite_ it (side a is opposite vertex A). - **Angles** — the three interior angles, written with the ∠ symbol: ∠A, ∠B, ∠C, or by three points as ∠BAC. This "lowercase side opposite capital vertex" rule is worth fixing early — it is assumed in every formula that follows. ## Types of Triangles Triangles are sorted two independent ways: by their **sides** and by their **angles**. Every triangle has one label from each list. ### By sides - [**Equilateral**](/content/math/geometry/equilateral-triangle/index.html) — all three sides congruent (equal), so all three angles equal 60°. - [**Isosceles**](/content/math/geometry/isosceles-triangles/index.html) — exactly two sides congruent, and the two angles opposite them equal. - [**Scalene**](/content/math/terms/scalene-triangle/index.html) — no sides equal, so all three angles differ. ### By angles - **Acute** — all three angles less than 90°. - **Right** — one angle exactly 90°. The side opposite the right angle is the hypotenuse. - **Obtuse** — one angle greater than 90°. See [obtuse triangles](/content/math/geometry/obtuse-triangles/index.html). The two labels combine. A triangle can be a **right isosceles** triangle (one right angle and two equal sides) or an **acute scalene** triangle (all angles under 90°, no sides equal), and so on. ## Properties of a Triangle These hold for _every_ triangle and are the rules you reach for to find missing measures. 1. **Angle sum property** - The three interior angles always add to 180°. ∠A + ∠B + ∠C = 180°. 2. **Triangle inequality** - The sum of any two sides is greater than the third side. If it weren't, the two shorter sides couldn't reach across to close the figure. a + b > c. 3. **Side opposite the larger angle is longer** - The longest side faces the largest angle; the shortest faces the smallest. 4. **Exterior angle property** - An exterior angle equals the sum of the two interior angles not next to it. \text{exterior angle} = \text{sum of the two remote interior angles}. 5. [**Congruence**](/content/math/geometry/congruence-in-triangles/index.html) **-** Two triangles are identical when they match by a congruence rule. A question students ask constantly: **can a triangle have two right angles?** No. Two right angles already use 90° + 90° = 180°, leaving 0° for the third — and a 0° angle is no angle at all, so the figure never closes. The angle-sum property rules it out directly. ## Area and Perimeter Formulas The **perimeter** is the distance around the triangle — just add the three sides: P = a + b + c. The **area** of a triangle is half the base times the height (the perpendicular distance from the base to the opposite vertex): Area = 1/2 ⋅ b ⋅ h. Here b is the chosen base and h is the height drawn perpendicular to that base. The 1/2 appears because a triangle is exactly half of the parallelogram you get by copying and flipping it — that is where the formula comes from, not a rule to memorize blindly. ## Examples of Triangles ### Example 1 **Two angles of a triangle are 75° and 60°. Find the third angle.** By the angle sum property, all three add to 180°. ∠C = 180° - 75° - 60° = 45°. **Final answer:** 45°. ### Example 2 **Can a triangle have sides 5 cm, 4 cm, and 9 cm?** A natural first move is to say "three lengths, so yes, it's a triangle." Test it against the triangle inequality. Take the two shorter sides: 5 + 4 = 9. The sum equals the third side exactly — it does not exceed it. Two sides laid end to end of total length 9 cm just barely reach across a 9 cm gap; they lie flat and never lift into a triangle. So these lengths give a flat, degenerate line, not a triangle. **Final answer:** No. Since 5 + 4 is not greater than 9, the inequality fails. ### Example 3 **Find the perimeter of a triangle with sides 3 cm, 4 cm, and 5 cm.** P = 3 + 4 + 5 = 12 cm. **Final answer:** 12 cm. (As a bonus, 3² + 4² = 5², so this is also a right triangle.) ### Example 4 **A triangle has a base of 10 cm and a height of 6 cm. Find its area.** Area = 1/2 ⋅ 10 ⋅ 6 = 30 cm². **Final answer:** 30 cm². ### Example 5 **One exterior angle of a triangle is 110°. One of the two remote interior angles is 45°. Find the other remote interior angle.** By the exterior angle property, the exterior angle equals the sum of the two remote interior angles. 110° = 45° + x. **Final answer:** x = 65°. ### Example 6 **A triangular sail has angles in the ratio 2:3:4. Classify the sail by its angles.** The angles add to 180°, and the ratio has 2 + 3 + 4 = 9 parts. one part = 180°/9 = 20°. So the angles are 40°, 60°, and 80°. Every angle is below 90°. **Final answer:** an acute triangle (and scalene, since all three angles — and so all three sides — differ). ## Why Triangles Are Everywhere: Rigidity And Reach The triangle's importance is not arbitrary. It is the _simplest_ closed straight-sided figure — you cannot make a closed shape from two straight sides — and it is the _only_ rigid one. - **Rigidity in structures.** Fix the three side lengths of a triangle and its shape is completely determined; there is no way to flex it without breaking a side. Quadrilaterals and larger polygons can deform. This is why engineers triangulate — bridge trusses, roof frames, cranes, and electricity pylons are webs of triangles. - **Reach in measurement.** Because a triangle is pinned down by the right combination of sides and angles, surveyors and astronomers use triangles to find distances they cannot measure directly — the basis of triangulation and the parallax method for measuring the distance to nearby stars. - **The foundation idea.** Every polygon can be cut into triangles. Computer graphics render curved surfaces as meshes of millions of tiny triangles for exactly this reason: the triangle is the atom of shape, the piece everything else is built from. Triangulation as a surveying method reaches back to Eratosthenes, who used angle measurements to estimate the size of the Earth around 240 BCE — a triangle's worth of geometry measuring a planet. ## Where Students Go Wrong With Triangles ### Mistake 1: Assuming any three lengths form a triangle **Where it slips in:** Given three side lengths, a student starts computing area or angles without checking whether the triangle can exist. **Don't do this:** Treat 2 cm, 3 cm, and 7 cm as a triangle and plug into a formula. **The correct way:** Apply the triangle inequality first — the two shorter sides must sum to _more_ than the longest. Here 2 + 3 = 5 < 7, so no triangle exists. The rusher skips this check and produces an "area" for a shape that cannot be drawn. ### Mistake 2: Confusing the height with a side **Where it slips in:** Using the slanted side as the height in the area formula instead of the perpendicular distance. **Don't do this:** Write Area = 1/2 ⋅ base ⋅ slant side. **The correct way:** The height is the _perpendicular_ drop from the opposite vertex to the base, not the length of a side (unless that side happens to be perpendicular, as in a right triangle). The memorizer who learned "half base times the other number" grabs whatever length is handy and overstates the area. ### Mistake 3: Forgetting a triangle carries two type-labels **Where it slips in:** A student names a triangle "isosceles" and stops, ignoring its angle type. **Don't do this:** Treat the side-type and angle-type lists as alternatives where you pick one. **The correct way:** Every triangle has both a side label (equilateral / isosceles / scalene) and an angle label (acute / right / obtuse). A full description gives both — "right isosceles," "acute scalene." The student who answers a "fully classify" question with one word leaves marks on the table. ## Key Takeaways - A **triangle** has three sides, three angles, and three vertices, with interior angles summing to 180°. - Triangles are classified by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse), one label from each. - The triangle inequality, angle-sum, exterior-angle, and side-opposite-larger-angle rules govern every triangle. - Perimeter is the sum of sides; area is 1/2 ⋅ base ⋅ height. - The triangle is the only rigid polygon, which is why it underlies structures, surveying, and computer graphics.