Inscribed circle in a right triangle
A circle sits inside a right-angled triangle, touching all three sides. Most questions give one of the shorter sides and the circle's radius and ask for the longest side; others ask for the other shorter side, or give the sides and ask for the radius. Some are plain diagrams, others a triangular garden with a round pond touching all three fences.
Two facts crack it. The radii to the two shorter sides make a square in the right-angle corner, so each shorter side has a piece as long as the radius by that corner. And the two lines from a corner to where the circle touches are always the same length. After that it is Pythagoras with one unknown piece, and the step-by-step working colours each equal pair on the picture.
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What changes each time
- Three forms: the longest side, the other shorter side, or the radius (from all three sides, or from two with Pythagoras first)
- Whole-number right triangles, from 3-4-5 to 65-72-97 and their multiples, drawn close to scale and either way round
- Longest side up to 75 in the diagrams (cm or bare numbers) and up to 100 m in the gardens; radius from 1 to 20
More shapes and angles practice
- Sides and corners of 2D shapes (Grade 2)
- Faces, edges and vertices (Grade 3)
- Sorting quadrilaterals (Grade 3)
- Turning and flipping letters (Grade 3)
- Counting squares of every size (Grade 3)
- Inside or outside the loop (Grade 3)
- Which piece fills the gap? (Grade 3)
- Symmetry on a grid (Grade 4)
- Rotations and reflections (Grade 4)
- Reading a protractor (Grade 4)
- Parallel and perpendicular lines (Grade 4)
- Parts of a circle (Grade 4)
- Draw without lifting your pencil (Grade 4)
- Can these sticks make a triangle? (Grade 4)
- 3D shapes: side view (Grade 5)
- Rectangle in a quarter circle (Grade 5)
- Painted cube puzzle (Grade 5)
- Rope around the Earth (Grade 5)
- Can dominoes cover the board? (Grade 5)
- Angles in a circle (Grade 6)