ShapesProperties
- Two sides are equal, and the angles opposite them are equal.
- It has one line of symmetry, through the top corner and the middle of the base.
- The height to the base cuts the base in half.
- An equilateral triangle is a special isosceles triangle.
Why the formulas work
The height from the top corner meets the base at its middle point. That makes a right-angled triangle with hypotenuse a and one leg b/2, so the height is the root of a² minus b²/4. With the height known, the area is half of base times height.
Worked example
Take a isosceles triangle with equal side 13 cm and base 10 cm.
- Height. Height = √(a2 − b2 ⁄ 4). Here: √(13² − 10² ⁄ 4) = 12 cm.
- Area. Area = ½ × b × height. Here: ½ × 10 × 12 = 60 cm².
- Perimeter. Perimeter = 2a + b. Here: 2 × 13 + 10 = 36 cm.
Finding a missing measurement
If the base and the height are known, the equal side is √(h² + b² ⁄ 4): a base of 10 cm and a height of 12 cm give √(144 + 25) = 13 cm. From the perimeter, the base is the perimeter minus twice the equal side.
Mistakes to avoid
- Taking the equal side as the height.
- Forgetting to halve the base before using Pythagoras.
- The two equal sides must together be longer than the base, or no triangle can be made.
Where you see it
A clothes hanger, the gable end of a house, a slice of pizza (nearly) and many roof trusses are isosceles triangles.
Try these
Work each one out, then tap to check.
A isosceles triangle has equal side 10 cm and base 12 cm. Find its height, area and perimeter.
Height = 8 cm; Area = 48 cm²; Perimeter = 32 cm.
A isosceles triangle has equal side 17 cm and base 16 cm. Find its height, area and perimeter.
Height = 15 cm; Area = 120 cm²; Perimeter = 50 cm.