# Heights, areas and solids · Years 5–6

> Heights, areas and solids, Years 5–6: Reasoning about area and more. Explanation, worked examples and free practice for the national curriculum in England.

In Heights, areas and solids (Years 5–6) you practise 5 skills: Heights of triangles and parallelograms · Area of a triangle and a parallelogram · Reasoning about area · Faces, edges and vertices · Prisms and pyramids.

The height to a side is a segment from the vertex opposite the side that meets the line of the side at a right angle. Every triangle has three heights, one for each side. In an obtuse triangle some heights lie outside the triangle and meet the extension of the side. In a parallelogram, a height joins a vertex to a side that does not meet it, at a right angle.

A triangle is half a rectangle: when a triangle and a rectangle have the same base and the triangle’s top vertex is on the rectangle’s top side, the area of the triangle is exactly half the area of the rectangle. You can cut a piece off a parallelogram and move it to make a rectangle with the same base and the same height. A diagonal cuts a parallelogram into two congruent triangles, so each triangle is half of it.

So the area of a parallelogram is the base times the height, and the area of a triangle is the base times the height, divided by 2: 6 × 4 ÷ 2 = 12. You can work out a triangle’s area from any side and the height to it, and it always comes out the same.

A prism has two identical bases facing each other, and its side faces are rectangles. A pyramid has one base, and its side faces are triangles that meet at one vertex. The base tells you everything: a prism with a pentagon for its base has 7 faces, 15 edges and 10 vertices, and a square pyramid has 5 faces, 8 edges and 5 vertices.

**Tip:** When you look for a height, look for the right angle: a height always meets the line of the side at a right angle, even when it is outside the shape.

## What you learn

### Heights of triangles and parallelograms

- Example: Which dashed segment is the height to the bottom side of the triangle? (A triangle and three dashed segments, marked A, B and C.)
- Options: A Segment A · B Segment B · C Segment C
- Answer: A
- Solution: Segment A starts at the vertex opposite the bottom side and meets its line at a right angle, so it is the height. Segment B is perpendicular to the side, but does not start at the vertex opposite it. Segment C is the height to another side. (The triangle and the height to the bottom side, with a right angle marked.)

### Area of a triangle and a parallelogram

- Example: The lengths in the picture are in cm. What is the area of the parallelogram, in cm²? (A parallelogram, with the dashed height to the bottom side and the lengths.)
- Answer: 30 cm²
- Solution: 10 × 3 = 30 The area of the parallelogram is 30 cm².

### Reasoning about area

- Example: The triangle and the rectangle have the same base. Is the area of the triangle more than half the area of the rectangle, less than half, or exactly half? (A rectangle on squared paper, with a triangle inside it whose base is the bottom side of the rectangle.)
- Options: More than half · Less than half · Exactly half
- Answer: Exactly half
- Solution: The top vertex of the triangle is on the top side of the rectangle, so the height of the triangle equals the height of the rectangle: 6. The area of the triangle, in squares: 4 × 6 ÷ 2 = 12 Half the area of the rectangle: 4 × 6 ÷ 2 = 12 The two results are equal: the area of the triangle is exactly half the area of the rectangle.

### Faces, edges and vertices

- Example: (square pyramid) How many faces does a square pyramid have?
- Answer: 5
- Solution: The base is a quadrilateral, and it has 4 sides. There are 4 triangular faces, plus the base: 4 + 1 = 5

### Prisms and pyramids

- Example: (A drawing of a solid.) Which polygon is the base of the solid?
- Options: quadrilateral · pentagon · triangle · hexagon
- Answer: triangle
- Solution: The base has 3 sides, so it is a triangle. The solid is a triangular pyramid.

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

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