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Home›Geometry and drawings

Triangles

Seven units on triangles: sorting them by sides and angles, the 180° angle sum and the exterior angle, which three lengths can make a triangle, the congruence conditions and why side-side-angle fails, similar triangles and scale, the Pythagorean theorem with two proofs and its converse, and area from base and height or from three sides with Heron’s formula. Drag the corners of a triangle in the trig lab and watch the numbers change.

Progress

About this course

Tear the three corners off any paper triangle and lay them point to point. They fill a straight line, 180°, whatever triangle you started with. That is the angle sum, and it is the first fact of the course. The trig lab rounds each angle to 0.1°, so the three rounded numbers can miss by 0.1, but the exact sum shown on the Sum line is 180°.

The seven units are types of triangles, the angle sum and the exterior angle, which three lengths make a triangle, congruent triangles (SSS, SAS, ASA and why SSA fails), similar triangles and scale, the Pythagorean theorem with two proofs and the converse, and area with Heron's formula. Two figures are similar when one is an enlargement or reduction of the other, so matching angles are equal and matching sides are all in the same ratio, called the scale factor. On squared paper, a right triangle with legs of 3 and 4 squares gets squares of 9 and 16 on its legs, and the square on its longest side covers exactly 25.

The last unit finds area two ways. The second route needs no height at all: for sides a, b and c, find s, half the perimeter, then Area = √(s(s − a)(s − b)(s − c)). For 13, 14 and 15, s = 42 ÷ 2 = 21, and the area is √(21 × 8 × 7 × 6) = √7056 = 84. Drag the corners of a triangle in the trig lab and watch the numbers change.

Before you start

Adding angles, comparing lengths and working with ratios help with the examples here. Angles is useful preparation for the straight-angle and parallel-line arguments. The triangle lab lets you move vertices and compare measurements, but the lessons also ask why a rule holds rather than only what the display shows.

When you finish

You can classify a triangle by sides and angles, find missing angles, and test whether three lengths can form a triangle. You can use congruence conditions, explain why SSA is insufficient, and calculate a height from similar triangles. You can apply the Pythagorean theorem and find area from a base and height or Heron’s formula.

  1. 1Types of triangles by sides and anglesA triangle with sides of 5, 5 and 8 cm is isosceles and obtuse at the same time. The first name comes from its sides, the second from its largest angle, about 106°.
  2. 2The angle sum and the exterior angleTear the three corners off any paper triangle and lay them point to point. They fill a straight line, 180°, whatever triangle you started with.
  3. 3Which three lengths make a triangleSticks of 3 cm, 4 cm and 8 cm cannot be joined into a triangle. The two short ones reach only 3 + 4 = 7 cm, one centimetre short of the long one.
  4. 4Congruent triangles: SSS, SAS, ASA and why SSA failsTwo triangles can share a 30° angle, a side of 10 and a side of 6 and still differ: the third side is about 11.98 in one and about 5.34 in the other.
  5. 5Similar triangles and scaleA 1.6 m student casts a 2 m shadow while the school flagpole casts one of 15 m. The pole is 1.6 × 15 ÷ 2 = 12 m tall, and nobody had to climb it.
  6. 6The Pythagorean theorem, two proofs and the converseOn squared paper, a right triangle with legs of 3 and 4 squares gets squares of 9 and 16 on its legs, and the square on its longest side covers exactly 25.
  7. 7Area of a triangle and Heron’s formulaA triangle with sides of 13, 14 and 15 m has an area of exactly 84 m², and you can reach that number by two quite different routes.

Related courses

  • Angles
    Five units on angles: measuring them with a protractor, naming the types, the pairs that add to 90° and 180°, the eight angles where a line crosses two parallel lines, and the angle sums of triangles and polygons. The trig lab on each page has a protractor, a pair of parallel lines and a triangle whose corners you can drag.
  • Area and volume of solids
    A cereal box, a tin can, an ice cream cone and a ball each have a formula for how much card covers them and how much they hold. Unfold each solid in 3D, change its size with sliders and watch the numbers follow.
  • Circles
    Six units on circles: the parts of a circle and how to find its centre, pi and the circumference with Archimedes’ polygon bounds, the area of a circle from rearranged sectors, arc length and sector area as fractions of 360°, central and inscribed angles with the angle in a semicircle, and tangents, where the radius meets the tangent at a right angle and two tangents from one point are equal. Drag a point round a circle, unroll a cylinder, unfold a cone and click the slices of a pie chart.
  • Trigonometry
    Six units of right-triangle and unit-circle trigonometry. You name the sides of a right triangle, define sine, cosine and tangent as ratios, learn the exact values for 30°, 45° and 60°, solve triangles from one side and one angle, measure heights you cannot reach, switch to radians and the unit circle, and draw the sine wave. The trig lab has a resizable right triangle, a height-measuring range and a unit circle tied to its graph.
Sources for this course (7)
  • Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 초등학교 3~4학년군 [4수03-08] 이등변삼각형·정삼각형, [4수03-09] 직각삼각형·예각삼각형·둔각삼각형, [4수03-25] 삼각형과 사각형의 내각의 크기의 합을 여러 가지 방법으로 추론; 초등학교 5~6학년군 [6수03-01] 합동·대응점·대응변·대응각, [6수03-14] 삼각형의 넓이(높이가 삼각형의 외부에 있는 것도 다룸); 중학교 1~3학년군 [9수03-03] 삼각형의 작도, [9수03-04] 삼각형의 합동 조건, [9수03-05] 삼각형의 내각과 외각, [9수03-09] 이등변삼각형의 성질을 합동 조건으로 정당화하고 증명이라는 용어 도입, [9수03-12]·[9수03-13] 닮음·닮음비·삼각형의 닮음 조건, [9수03-15] 피타고라스 정리를 추측하고 정당화, 역은 직관적으로 이해하고 직각삼각형 판별에 사용; 용어와 기호 대변, 대각, 내각, 외각, 증명, 닮음비, △ABC, ≡, ∽)
  • Textbook: OpenStax, Contemporary Mathematics, 10.3 Triangles (triangle from three noncollinear points; acute, right, obtuse; side a opposite angle A; hypotenuse and legs; angle sum 180° by a parallel line and alternate interior angles; x, x + 17, 3x − 62 gives 45°, 62°, 73°; SSS, SAS, ASA, AAS; similar triangles, scaling factor; tower shadow 180 ft with a 9 ft truck and 15 ft shadow gives 108 ft; Thales and the pyramid: openstax.org/books/contemporary-mathematics/pages/10-3-triangles)
  • OpenStax Prealgebra 2e, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²; Pythagoras around 500 BCE; 13 ft ladder 5 ft from the wall; 10 inch bracket about 7.1 inches; similar triangles: openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem)
  • 9.4 Use Properties of Rectangles, Triangles, and Trapezoids (A = ½bh from half a rectangle; isosceles and equilateral; equilateral perimeter 93 in: .../pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids)
  • OpenStax Precalculus 2e, 8.2 Non-right Triangles: Law of Cosines (Heron’s formula with semi-perimeter s; sides 10, 15, 7 give s = 16 and area about 29.4; Chicago lot 62.4, 43.5, 34.1 m gives about 711.4 m²: openstax.org/books/precalculus-2e/pages/8-2-non-right-triangles-law-of-cosines)
  • History for story blocks only: Euclid’s Elements, D. E. Joyce edition, Clark University (Book I Prop. 20 triangle inequality and Heron’s shortest-path problem; Prop. 26 and its guide on congruence theorems, SSA ambiguity and side-side-right angle; Prop. 32 exterior angle and angle sum, Proclus corollaries; Prop. 47 and its guide on Old Babylonians ca. 1900 to 1600 BCE, Plimpton 322, the Zhou bi hypotenuse diagram, Proclus; Prop. 48 converse; Book VI Prop. 4 equiangular triangles: mathcs.clarku.edu/~djoyce/elements/bookI/propI47.html)
  • MacTutor History of Mathematics, University of St Andrews, Heron of Alexandria (formula proved in Metrica Book I; eclipse of 13 March 62; side of 720 about 26 5/6: mathshistory.st-andrews.ac.uk/Biographies/Heron/) MacTutor, "Euclid of Alexandria" (born about 325 BC, died about 265 BC; dating for the Elements, about 2,300 years ago: mathshistory.st-andrews.ac.uk/Biographies/Euclid/, checked 2026-10-19).

Checked: 2026-10-19